1. Introduction

Optical sorting is based on the force interaction between a laser beam and an object mediated by the light scattered by the object. Two principally different approaches have been applied in the past: active and passive sorting [1–3]. Active sorting usually uses some kind of detection and analysis of object properties using CCD image [4–7], scattering pattern [8], fluorescence [7, 9–11], or spectroscopy [12–15]. Based on these properties the optical force is applied so as to move the sorted object to collection or waste areas. This type of sorting is usually used in combination with a fluid flow in microfluidic systems, where the optical force deflects the sorted object into the streamline pointing to the collection or waste channel [5, 7, 11, 16]. It can be part of more complex lab-on-a-chip setups [15, 17]. In contrast, passive sorting exploits the dependence of the optical force on particle properties to vary particle trajectories [18–29] or modify their speed along the fluid flow (optical chromatography) [30–33]. Some of these methods can be even applied in a static fluid. The sorting is then based on the size dependent radiation pressure applied in opposite directions [34,35], the principle of space-time symmetry breaking in optical ratchets [36–43] or the travelling interference pattern, usually periodic, that drags particles of different sizes or refractive indices with different strength [44–48]. The latter method is based on the nonlinear mobility of particles when these are forced to move in a sinusoidal energy potential [49].

When the fringes move, an effective tilted energy potential appears in the travelling fringe reference frame (see Fig. 1). Depending on the speed of the fringes, the energy barriers can be diminished, increased or even removed, giving rise to significant difference in the particle dynamics [50]. If the energy barrier is much larger than the thermal energy ΔU ≫ k_BT, where k_B and T denote the Boltzmann constant and absolute temperature, respectively, the particle can be kept at the same potential minimum (interference fringe) following the fringe motion deterministically (the left column in Fig. 1). However, if ΔU ≃ k_BT, the particle may surmount the potential barrier due to the thermal activation, jump to the neighbouring fringe and lag behind the fringe motion (the right column in Fig. 1). In this paper we demonstrate experimentally several methods that use this principle to quickly sort spherical particles of different sizes and especially to separate living algal cells of different properties or spherical from nonspherical particles.

 

Fig. 1 Particle behaviour in travelling interference fringes. The blue particle feels deeper potential profile in the direction perpendicular to the fringes and thus if the fringes move to the right with velocity u_x, the particle stays in the same potential well but the Stokes drag force F_d = −γu_x pushes the particle to the left. Such a particle is referred to as a Brownian surfer [50]. This situation is identical to the tilted periodic potential in the travelling reference frame where the lower potential barrier is still height enough to keep the particle in the same fringe (for reasonably long time). In contrast, the red particle in the right column feels much shallower or no periodic potential and it lags behind the fringe or falls down to the left in the tilted potential picture. Such a particle is referred to as a Brownian swimmer [50].

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2. Particle behaviour in a travelling periodic potential: Physical background

Let us assume that the interference fringes shown in Fig. 1 are formed in xy plane due to the interference of two very wide identical Gaussian beams that can be considered for simplicity as plane waves of the intensity I₀, corresponding to the intensity at the Gaussiam beam waist, and vacuum wavelengths λ₀ but with wavevectors forming an angle β (see also experimental setup in Fig. 4). The xy plane is perpendicular to the resulting field wavevector k = k₁ + k₂ and the fringes are assumed to be parallel with y and z axes. The optical intensity can be written as

For a spherical particle of radius a and refractive index n_p in a medium of refractive index n_m close to n_p (i.e. n_p/n_m ≃ 1) the optical force F_x in the given field of Eq. (1) can be expressed analytically as [51]:

For the force profile given by Eqs. (2), the following periodic optical potential profile can be obtained along x axis [51]:

Since the microscopic particle is immersed in water, its motion is overdamped and it can be described by the Langevin equation in the form [52]:

The potential profile U_x can be transformed to the travelling wave frame x̄ = x − u_xt to give the static ‘tilted washboard potential’ shown in Fig. 1 in the form

In between those limiting cases, the potential well exists but its depth is comparable to k_BT and the particle eventually jumps to one of the neighbouring wells, preferentially down the potential hill [53, 54]. The mean time needed to overcome the barrier is called the ‘mean first passage time’ or just the Kramers time [55, 56]. The particle still follows the movement of the fringe but its mean velocity v_x is lower comparing to the fringe velocity u_x because of these jumps to the neighbouring potential wells. In the case of a travelling periodic potential, as considered here, the following relation is valid [50]:

As Fig. 2 reveals, this function has one global maximum giving maximal particle speed v_max for certain value of the fringe velocity u_max and the rest of the parameters fixed [57].

 

Fig. 2 An example of the dependence of the mean particle velocity v_x on the fringes travelling speeds u_x and the potential barrier heights ΔU. The polystyrene particle (radius 1 μm) is immersed in water (viscosity ν = 0.001 Pa s) far from any solid interface and illuminated with an interference field with a spatial period L = 1 μm. Green curve denotes the maximal particle velocity v_max for each trap depth. Regions on the left (u_x > 0) or the far right of the green curve correspond to the Brownian surfers or the Brownian swimmers, respectively.

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3. Particle sorting using the travelling periodic potential: Experimental results

3.1. Principle of the sorting methods

The methods directly use the conclusions coming from Eq. (11) and employ the fact that in our experiments we can tune the fringe separation L and their velocity u_x. If we find a combination of values L and u_x that provides fast particle velocity v_x for one particular particle size (see the blue particle in Fig. 3(a) and minimal velocity for the other particle size (see the red particle in Fig. 3(a), both sizes of the particles will be separated if the fringes move in one direction for a sufficiently long time (much longer than the mean first passage time the particle needs to overcome the barrier [55]). This method is a modification of the principle of optical chromatography [32, 58], but the fluid is static here and the components of the suspension are dragged in the fluid with different velocities.

 

Fig. 3 Description of the sorting procedure for spheres of two different diameters. a) The fringes (green) move to the left and drag faster the larger blue particle. Therefore the smaller red particle lags behind the blue one. b) The direction of the fringes movement is reversed with smaller distance L between fringes. The smaller particle is dragged faster than the bigger one. Repetition of this procedure several times leads to increasing distance between larger and smaller particles - components of a heterogeneous suspension.

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After a certain time (longer than the first mean passage time across the potential barrier) the direction of the fringes movement can be reversed and the values of L and u_x are modified in such a way that the behaviour of particles of both sizes is reversed. The previously fast particle is moves slowly (see the blue particle in Fig. 3(b) and the previously slow particle moves fast (see the red particle in Fig. 3(b). Repetitive changes of the motion direction, speed and period of the fringes lead to the separation of particles of different sizes and consequently to both components of the static heterogeneous suspension.

 

Fig. 4 Experimental setup (see the text for details).

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The proper parameters of the fringes for spheres of radii from 5 nm to 5 μm can be found from the extensive theoretical results presented in Figs. 9, 10, 11 of the Appendix for four considered refractive indices: close to water (n_p =1.344), silica (n_p =1.46), polystyrene (n_p =1.59), and melamine (n_p =1.68). These results related to polystyrene and denoted by white dashed or dotted lines in the figures were used for the experiments presented below. The dynamics expressed by Eq. (11) considers only a single particle or a diluted sample of particles. When two or more particles are close to each other, many complex physical effects such as optical binding, hydrodynamic or electrostatic interactions, or even jamming or percolation phenomena may be present and disturb the intended way of optical sorting according to Eq. (11). However, in our experiments presented below, they did not influence significantly the sorting conditions and we were able to sort samples which consisted of many particles.

3.2. Experimental setup

The experimental setup is shown in Fig. 4. The key element is the spatial light modulator (SLM) which provides dynamic modification of the phase mask and fringe separation L. The Gaussian beam coming out of the laser (Verdi V10, Coherent, λ₀=532 nm, maximal output power P_max = 10 W, output beam diameter d = 2.25 mm) is transformed by a telescope 1:10 (Thorlabs AC127-019-A, f=19 mm; AC254-200-A, f=200 mm) with a spatial filtering pinhole d = 10 μm in the focal plane (Thorlabs P10C). The enlarged beam is diffracted by a phase grating on the SLM. Its 0^th diffraction order is blocked in the focal plane of the long-focal-distance lens (Thorlabs AC508-750-A-ML, f=750 mm). The beams diffracted into the ±1^st diffraction orders create two focal points displaced symmetrically to the optical axis. Two different SLMs and slightly modified setups were used in the course of experiments to demonstrate the robustness of the method. The first setup (further referenced as ‘Configuration A’) used optically addressed high resolution SLM (Hamamatsu PAL-SLM, X8267-5080DB) coupled into the microscope objective (Olympus, Plan 40×, NA 0.65, air) with a telescope f(L1)=60 mm and f(L2)=200 mm (Thorlabs AC254-60-A and AC254-200-A), which projects those foci onto the back-focal-plane (BFP) of the microscope objective. The other configuration (‘Configuration B’) is suitable for rapid movement of interference fringes with a 10× faster SLM (Boulder HSPDM512-0532-PCI) combined with the telescope f(L1)=500 mm and f(L2)=250 mm (Thorlabs AC254-500-A and AC254-250-A) and an oil immersion microscope objective (Olympus, PlanC 100×, NA 1.25, oil). Finally, after passing through the objective, two collimated beams interfere in the sample space and create a 1D interference fringe pattern. The orientation and periodicity of these fringes can be tuned by the positions and distance between the focal points in BFP. Fringe movement is controlled by a phase shift added continuously in time to one of the beams. The speed of phase shift determines the velocity of fringe movement u_x. In both configurations (A and B), the active area in the sample is largely comparable (35×35 μm). The sample itself is placed between the cover slip and the bottom slide separated by a distance defined with spacer spheres (15 μm in diameter, Duke Scientific 4K-15). However, during the sorting the particles in the sample are pushed by the incident beams against the top surface.

3.3. Separation of spheres of different sizes

When we designed the experiments presented bellow, we first used the periods of fringes L predicted by the theoretical results presented in the Appendix. During the experiment we slightly tuned the selected period of fringes to get faster or slower motion of sorted objects. The final parameters are denoted by white dashed and dotted lines in Figs. 9, 10, 11 of the Appendix.

Figure 5 shows that using fast movement of the fringes across the suspension in one direction, two types of particles are dragged with different velocities and, after a time longer than the first passage time, they can be separated. Fringes are oriented horizontally and move vertically in the picture. Figure 6 shows the separation of two types of polystyrene microspheres of different sizes using repetitive motion of fringes upwards and downwards. Within 45 s of their movement in both directions the positions of both sizes are interchanged. At the beginning, the smaller spheres were above the bigger ones and during sorting they moved below them. The trajectories reveal that both sizes move with the fringes but with slightly different velocities which results in their separation after 4 cycles. The attached movie demonstrates that even the presence of many illuminated particles does not disturb significantly the intended motion of the sorted particles.

 

Fig. 5 An example of optical sorting of polystyrene spheres of 2 μm and 3 μm in diameter (Duke Scientific 4K-02 and 4K-03) using fast travelling (u_x = 14 μm.s⁻¹) fringes of periodicity L = 2.15 μm only in one direction. Left and right images show the starting and final distributions of the spheres, respectively. The middle part denotes the trajectories of all spheres during sorting obtained from Media 1. Configuration B was used here.

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Fig. 6 An example of optical sorting of polystyrene spheres of 0.8 μm and 1.6 μm in diameter (Duke Scientific 3K-800 and 3K1600) using bidirectional movement of the fringes of periods L = 0.66μm (travelling downwards in the figure) and L = 0.28μm (travelling upwards), respectively. Left and right images show the starting and final distributions of the spheres, respectively. The middle part denotes the trajectories of all spheres during sorting obtained from Media 2. Configuration A was used in this experiment.

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3.4. Separation of living algal cells

Figure 7 demonstrates separation of three algal cells Trachydiscus minutus of different sizes and compositions. A slightly different technique was used here. We applied cyclic movement of the sample located on XY piezo-stage (homemade device with range 50×50 μm) and at the turning points of the stage we changed the fringe separation so that smaller (larger) cells were held in the fringes (horizontal trajectories in the figure) while the larger (smaller) ones travelled almost freely with the sample (tilted trajectories in the figure). We found the period of fringes providing separation of living cells by tuning the starting fringe period that was estimated from the results for a spherical particle of refractive index close to water, till different algae moved differently in the fringes.

 

Fig. 7 An example of separation of living cells of Trachydiscus minutus using sample travelling in xy plane perpendicularly through static interference fringes. Left and right images show the starting and final distributions of cells, respectively. The middle part denotes the trajectory of each cell during sorting obtained from the Media 3. The periods of fringes were L = 3.5μm and L = 0.8μm in configuration A.

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3.5. Separation of spheres and spheroids of the same volume

Figure 8 illustrates the possibility to sort even objects of different shapes but almost equal volume. Spheroids were manufactured from polystyrene spheres [59–61] and after the procedure they were mixed with the original spheres and diluted by water. Travelling fringes were used to separate spheres and spheroids. Oblate spheroids were oriented with their minor axis towards the observer and kept this orientation for the fringe widths used for sorting purposes. In wide fringes they rotated by 90° as illustrated in the rectangular inset added to the right bottom part of Fig. 8. The preferred orientation and forces acting upon oblate spheroids in interference fringes are analysed in the Appendix.

 

Fig. 8 An example of optical sorting of polystyrene spheres (diameter 1.99 μm) and oblate spheroidal particles of equal volume and ratio of semi-axes close to 1.6. The initial position is shown on the left. Sharper smaller objects are spheres, larger ones are oblate spheroids oriented with their minor axis towards the observer. This orientation was preferred for the fringes widths used in this figure, L = 1540 nm (L = 990 nm) was used for spheroids (spheres) motion up (down). The right part demonstrates the final situation, both spheroids are sorted upwards. The inset added to the right bottom shows the sphere and the spheroids rotated by 90° if the fringe width increased to L > 2μm. Configuration B was used in this experiment. Trajectories were obtained from Media 4 and rotated by 90°.

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Appendix

Particle behaviour in the travelling periodic potential: Numerical results

The theory presented in Section 2 is valid only for particles of refractive index close to the refractive of the surrounding medium (in Figs. 9, 10, 11 it corresponds to n_p = 1.334, close to water). For larger refractive indices of experimental particles one has to use more complex approach. We used generalised Lorenz-Mie scattering theory [62] in Barton’s formalism [63] to calculate optical forces for several refractive indices of spherical particles n_p as a function of their radii a and distance between neighbouring fringe maxima L in order to obtain numerical results more relevant to the experiments. Figures 9, 10, 11 can be used to design the experimental parameters for wide range of particle sizes made of silica, polystyrene and melanin. Since the optical force F_x depends on the trapping wavelength only weakly through the dispersion of refractive indices n_p and n_m, Figs. 9, 10, 11 can be also used as the first estimate for different trapping wavelengths than considered 532 nm, although both particle sizes and fringe periods have to be scaled accordingly.

 

Fig. 9 Amplitude of the optical force $F_x^{\mathit{max}}(a,L)$ acting upon a particle of radius a placed into the interference fringes separated by L. The scaling of L along vertical axis is changed at L = 0.5 μm (marked by a dashed magenta line) in order to visualise area for L < 0.5 μm. White contours denote combination of parameters giving zero force amplitude. The selection of refractive index of the particles corresponds to one close to water (n_p =1.344), silica (n_p =1.46), polystyrene (n_p =1.59), and melamine (n_p =1.68). The sizes of polystyrene particles and fringe periods used in the experiments presented in Fig. 5 or Fig. 6 are marked with white dashed or dotted lines, respectively.

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Fig. 10 The trap depth ΔU (defined in Eq. (4)) for a particle of radius a placed into the interference fringes separated by L. The trap depth is shown in k_BT units assuming T = 300 K. The used symbolism is the same as in Fig. 9. The scaling of the colour maps is split by the solid magenta curve into two regions in order to increase readability of lower trap depths.

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Fig. 11 Maximal average speed of particle v_max (the left column) is achieved for the fringes speed u_max ploted in the right column. These speeds were calculated by finding global maximum of Eq. (11) using the trap depth ΔU taken from Fig. 10. The used symbolism is the same as in Fig. 9.

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Figure 9 shows amplitude $F_x^{\mathit{max}}(a,L)$ of optical force along x axis acting upon a particle of radii a =5 nm − 5 μm, fringe separation L = 0.2 − 3.0 μm and particle refractive indices n_p =1.344 (close to n_m), 1.46 (silica), 1.59 (polystyrene), and 1.68 (melamine). We have considered that optical intensity of each interfering plane wave corresponds to 1 W passing through the circular aperture of diameter 50 μm.

Figure 10 shows the trap depth ΔU calculated from the data presented in Fig. 9. The region between a ≃ 0 and the first white contour corresponds to the particle equilibrium position at the fringe intensity maximum. The next neighbouring region between white contours corresponds to the particle equilibrium position at the fringe intensity minimum. Similarly in the consequent regions, the particle position changes between fringe antinodes and nodes. Further, we have found numerically both v_max and u_max giving maximal values of v_x in Eq. (11) for all previously calculated parameters values. These results are summarised in Fig. 11 where the left (right) column shows v_max (u_max) and each row corresponds to different particle refractive index.

Separation of spheres and oblate spheroids

We used ‘T-matrix’ approach [64] to calculate optical forces and torques acting upon spheroids. Figure 12 presents results associated with the experimental sorting of polystyrene spheres and oblate spheroids. We plot the amplitude of the force efficiency $Q_x^F$ which is related to $F_x^{\text{max}}$ as

(12)$$F_x^{\mathit{max}}=\frac{I_0{\lambda }_{\mathit{vac}}^2}{4{\pi }^2{n}_{m}c}{Q}_x^F.$$

Plots in each column correspond to one of three distinct orientations of the oblate spheroid short symmetry axis c with respect to the interference fringes. The red curves correspond to the spheroids of selected aspect ratios a_⊥/a_‖ = 1.2, 1.4 and 1.6. The optical force acting on a spherical object is included in each plot in black colour for comparison.

 

Fig. 12 Top row: geometry of an oblate spheroid orientation in the interference fringes (parallel with yz plane). Plots below: Comparison of amplitude of the force efficiency $Q_{x,0}^F$ as a function of the fringe period, spheroid orientations and aspect ratios. We considered sphere of diameter 1990 nm (black curve) and spheroid of the same volume (red curve). The thick parts of the red curves denote intervals in L where the theory predicts stable orientation. The green bars denote configuration (a_⊥/a_‖ = 1.6) used in the experiments shown in Fig. 8.

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Thick red parts of the curves denote such fringe periods L where the spheroid orientation is stable (i.e. each of the two nonzero torque derivatives are negative: ∂T_z/∂ϕ_z, ∂T_y/∂ϕ_y for the case of c ‖ x; ∂T_z/∂ϕ_z, ∂T_x/∂ϕ_x for c ‖ y; ∂T_y/∂ϕ_y, ∂T_x/∂ϕ_x for c ‖ z). These criteria assess the simplest case when the spheroid centre is fixed in free space. However in contrast to the experimental observations (with prevailing spheroid orientation c ‖ z), the plots for c ‖ z reveal instability for all assumed L. We attribute this discrepancy mainly to the hydrodynamical and optical interaction with the surface which is difficult to address in the theory. Apparently in the last orientation c ‖ z, the spheroid centre is closer to the surface and the omitted hydrodynamical and optical interactions with the surface can significantly influence the spheroid stability and dynamics. The green lines for μ = a_⊥/a_‖=1.6 denote fringe periods L₁=990 nm and L₂=1540 nm used in the experimental sorting described above. We obtain qualitative coincidence with the experiment, fringes with period L₁=990 nm (L₂=1540 nm) ensure larger force on spheres (spheroids).

Due to different shapes of both sorted particles, the differences in hydrodynamic friction coefficients should be considered to compare the velocities of both sorted particle shapes [65]. If the oblate spheroid of aspect ratio μ = a_⊥/a_‖ > 1 is dragged through the fluid of viscosity η at a uniform velocity v parallel/perpendicular to the spheroid axis, the force F = vf_‖,⊥ is proportional to the velocity magnitude v via the Perrin’s factors:

(13)$$f_{\Vert }=6\pi \eta a\frac{4}{3}{\mu }^{-2/3}\frac{1-{\mu }^2}{(2-{\mu }^2)S(\mu )-1},$$

(14)$$f_{\perp }=6\pi \eta a\frac{8}{3}{\mu }^{-2/3}\frac{1-{\mu }^2}{(2-3{\mu }^2)S(\mu )+1},$$

(15)$$\text{where}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}S(\mu )=\frac{{\text{tan}}^{-1}\sqrt{{\mu }^2-1}}{\sqrt{{\mu }^2-1}}.$$

Figure 13 compares Perrin’s factors for both orientations of the oblate spheroid. The results show that for μ = 1.6 the hydrodynamic friction for orientation c ‖ z is about 1% lower comparing to a sphere. Therefore, the spheroid shape will not contribute significantly to the sorting velocities. Stronger hydrodynamic influence would come from the surface proximity, however detailed study of this influence is out of the scope of this paper.

 

Fig. 13 Dependence of Perrin’s translation friction coefficients Eqs. (13, 14) on the aspect ratio of the oblate spheroid. The full black line denotes the constant value for sphere.

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4. Conclusions

This paper presents three methods of the passive optical sorting of microobjects in a static suspension using interference fringes. The first method uses one fast movement of the fringes across the suspension to separate spherical particles of different sizes. The second method uses a bidirectional repetitive fringe motion with different fringe distance in each direction to achieve separation of spherical particles of different sizes or spherical from nonspherical particles. The third method, applied on sorting of living algal cells, is based on a bidirectional motion of the microscope stage with different fringe distances for each stage direction. The presented results and attached movies show that the suspension components within the field of view were separated in less than one minute. We find very good coincidence between the numerical results and experiments and we use the theoretical model successfully to set the sorting parameters in the experiments.

Acknowledgments

The authors highly appreciate suggestions of Dr. Stephen Simpson. The research was supported by Ministry of Education, Youth and Sports of the Czech Republic (MEYS) (projects LD14069, LH12018, LO1212) and CONACYT-ASCR Mexico-Czech Republic bilateral cooperation project (grant 171478). The research infrastructure was supported by MEYS, ASCR, and EC (projects LO1212, RVO: 68081731, and CZ.1.05/2.1.00/01.0017).

References and links

1. B. S. Zhao, Y.-M. Koo, and D. S. Chung, “Separations based on the mechanical forces of light,” Anal. Chim. Acta 556, 97–103 (2006). [CrossRef]  

2. K. Dholakia, M. P. MacDonald, P. Zemánek, and T. Čižmár, “Cellular and colloidal separation using optical forces,” Methods Cell Biol. 82, 467–495 (2007). [CrossRef]   [PubMed]  

3. A. Jonáš and P. Zemánek, “Light at work: The use of optical forces for particle manipulation, sorting, and analysis,” Electophoresis 29, 4813–4851 (2008). [CrossRef]  

4. S. C. Chapin, V. Germain, and E. R. Dufresne, “Automater trapping, assembly, and sorting with holographic optical tweezers,” Opt. Express 14, 13095–13100 (2006). [CrossRef]   [PubMed]  

5. J. Kovac and J. Voldman, “Intuitive, image-based cell sorting using optofluidic cell sorting,” Anal. Chem. 79, 9321–9330 (2007). [CrossRef]   [PubMed]  

6. I. R. Perch-Nielsen, D. Palima, J. S. Dam, and J. Glückstad, “Parallel particle identification and separation for active optical sorting,” J. Opt. A 11, 034013 (2009). [CrossRef]  

7. X. Wang, S. Chen, M. Kong, Z. Wang, K. Costa, R. Li, and D. Sun, “Enhanced cell sorting and manipulation with combined optical tweezer and microfluidic chip technologies,” Lab Chip 11, 3656–3662 (2011). [CrossRef]   [PubMed]  

8. T. N. Buican, M. J. Smyth, H. A. Crissman, G. C. Salzman, C. C. Stewart, and J. Martin, “Automated single-cell manipulation and sorting by light trapping,” Appl. Opt. 26, 5311–5316 (1987). [CrossRef]   [PubMed]  

9. M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, “Microfluidic sorting of mammalian cells by optical force switching,” Nat. Biotechnol. 23, 83–87 (2005). [CrossRef]  

10. H. I. Kirei, L. Oroszi, S. Valkai, and P. Ormos, “An all optical microfluidic sorter,” Acta Biologica Hungarica 58, 139–148 (2007). [CrossRef]  

11. T. Perroud, J. Kaiser, J. Sy, T. Lane, C. Branda, A. Singh, and K. Patel, “Microfluidic-based cell sorting of Francisella tularensis infected macrophages using optical forces,” Anal. Chem. 80, 6365–6372 (2008). [CrossRef]   [PubMed]  

12. C. Xie, D. Chen, and Y. Li, “Raman sorting and identification of single living micro-organisms with optical tweezers,” Opt. Lett. 30, 1800–1802 (2005). [CrossRef]   [PubMed]  

13. A. C. De Luca, S. Manago, M. A. Ferrara, I. Rendina, L. Sirleto, R. Puglisi, D. Balduzzi, A. Galli, P. Ferraro, and G. Coppola, “Non-invasive sex assessment in bovine semen by Raman spectroscopy,” Laser Phys. Lett. 11, 055604 (2014). [CrossRef]  

14. O. Samek, P. Zemánek, A. Jonáš, and H. H. Telle, “Characterization of oil-producing microalgae using Raman spectroscopy,” Laser Phys. Lett. 8, 701–709 (2011). [CrossRef]  

15. S. Dochow, C. Krafft, U. Neugenbauer, T. Bocklitz, T. Henkel, G. Mayer, J. Albert, and J. Popp, “Tumour cell identification by means of Raman spectroscopy in combination with optical traps and microfluidic environments,” Lab Chip 11, 1484–1490 (2011). [CrossRef]   [PubMed]  

16. B. Landenberger, H. Höfemann, S. Wadle, and A. Rohrbach, “Microfluidic sorting of arbitrary cells with dynamic optical tweezers,” Lab Chip 12, 3177–3183 (2012). [CrossRef]   [PubMed]  

17. X. Wang, X. Gou, X. Yan, and D. Sun, “Cell manipulation tool with combined microwell array and optical tweezers for cell isolation and deposition,” J. Micromech.Microeng. 23, 75006 (2013). [CrossRef]  

18. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003). [CrossRef]   [PubMed]  

19. K. Ladavac, K. Kasza, and D. G. Grier, “Sorting mesoscopic objects with periodic potential landscapes: Optical fractionation,” Phys. Rev. E 70, 010901 (2004). [CrossRef]  

20. M. Pelton, K. Ladavac, and D. G. Grier, “Transport and fractionation in periodic potential-energy landscapes,” Phys. Rev. E 70, 031108 (2004). [CrossRef]  

21. A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94, 160601 (2005). [CrossRef]   [PubMed]  

22. R. W. Applegate Jr., J. Squier, T. Vestad, J. Oakey, D. W. M. Marr, P. Bado, M. A. Dugan, and A. A. Said, “Microfluidic sorting system based on optical waveguide integration and diode bar trapping,” Lab Chip 6, 422–426 (2006). [CrossRef]   [PubMed]  

23. X. Yuan, S. Zhu, J. Bu, Y. Sun, J. Lin, and B. Gao, “Large-angular separation of particles induced by cascaded deflection angles in optical sorting,” Appl. Phys. Lett. 93, 263901 (2008). [CrossRef]  

24. R. F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P. J. Reece, T. F. Krauss, M. Gu, and K. Dholakia, “Optical deflection and sorting of microparticles in a near-field optical geometry,” Opt. Express 16, 3712–3726 (2008). [CrossRef]   [PubMed]  

25. G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optically generated potential energy landscapes,” Opt. Lett. 32, 1144–1146 (2007). [CrossRef]   [PubMed]  

26. S. Ahlawat, R. Dasgupta, R. Verma, V. Kumar, and P. Gupta, “Optical sorting in holographic trap arrays by tuning the inter-trap separation,” J. Opt. 14, 125501 (2012). [CrossRef]  

27. H. Xin, D. Bao, F. Zhong, and B. Li, “Photophoretic separation of particles using two tapered optical fibers,” Laser Phys. Lett. 10, 036004 (2013). [CrossRef]  

28. X. Wu, Y. Zhang, C. Min, S. Zhu, J. Feng, and X.-C. Yuan, “Dynamic optical tweezers based assay for monitoring early drug resistance,” Laser Phys. Lett. 10, 065604 (2013). [CrossRef]  

29. G. Volpe, G. Volpe, and S. Gigan, “Microfluidic sorting with blinking optical traps,” Sci. Rep. 4, 3936 (2014).

30. S. J. Hart and A. V. Terray, “Refractive-index-driven separation of colloidal polymer particles using optical chromography,” Appl. Phys. Lett. 83, 5316–5318 (2003). [CrossRef]  

31. S. J. Hart, A. Terray, T. Leski, J. Arnold, and R. Stroud, “Discovery of a significant optical chromatographic difference between spores of Bacillus anthracis and its close relative, Bacillus thuringiensis,” Anal. Chem. 78, 3221–3225 (2006). [CrossRef]   [PubMed]  

32. A. Terray, J. Taylor, and S. Hart, “Cascade optical chromatography for sample fractionation,” Biomicrofluidics 3, 044106(2009). [CrossRef]  

33. P. Ashok, R. Marchington, P. Mthunzi, T. Krauss, and K. Dholakia, “Optical chromatography using a photonic crystal fiber with on-chip fluorescence excitation,” Opt. Express 18, 6396–6407 (2010). [CrossRef]   [PubMed]  

34. P. Jákl, T. Čižmár, M. Šerý, and P. Zemánek, “Static optical sorting in a laser interference field,” Appl. Phys. Lett. 92, 161110 (2008). [CrossRef]  

35. O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čizmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ‘tractor beam’,” Nat. Photonics 7, 123–127 (2013). [CrossRef]  

36. B. A. Koss and D. G. Grier, “Optical peristalsis,” Appl. Phys. Lett. 82, 3985–3987 (2003). [CrossRef]  

37. A. Libál, C. Reichhardt, B. Jankó, and C. J. O. Reichhardt, “Dynamics, rectification, and fractionation for colloids on flashing substrates,” Phys. Rev. Lett. 96, 188301 (2006). [CrossRef]   [PubMed]  

38. R. Dasgupta, R. Verma, and P. Gupta, “Microfluidic sorting with blinking optical traps,” Opt. Lett. 37, 1739–1741 (2012). [CrossRef]   [PubMed]  

39. A. Neild, T. Ng, and T. Woods, “Optimizing photophoresis and asymmetric force fields for grading of Brownian particles,” Appl. Opt. 48, 6820–6826 (2009). [CrossRef]   [PubMed]  

40. Y. Hayashi, S. Ashihara, T. Shimura, and K. Kuroda, “Simultaneous separation of polydisperse particles using an asymmetric nonperiodic optical stripe pattern,” Appl. Optics 48, 1543–1552 (2009). [CrossRef]  

41. Y. Hayashi, S. Ashihara, T. Shimura, and K. Kuroda, “Particle sorting using optically induced asymmetric double-well potential,” Opt. Commun. 281, 3792–3798 (2008). [CrossRef]  

42. A. V. Arzola, K. Volke-Sepúlveda, and J. L. Mateos, “Dynamical analysis of an optical rocking ratchet: Theory and experiment,” Phys. Rev. E 87, 062910(2013).

43. A. V. Arzola, K. Volke-Sepúlveda, and J. L. Mateos, “Experimental control of transport and current reversals in a deterministic optical rocking ratchet,” Phys. Rev. Lett. 106, 168104 (2011). [CrossRef]  

44. T. Čižmár, M. Šiler, M. Šerý, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, “Optical sorting and detection of sub-micron objects in a motional standing wave,” Phys. Rev. B 74, 035105 (2006). [CrossRef]  

45. I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, “Modulated optical sieve for sorting of polydisperse microparticles,” Appl. Phys. Lett. 88, 121116(2006). [CrossRef]  

46. R. L. Smith, G. C. Spalding, K. Dholakia, and M. P. MacDonald, “Colloidal sorting in dynamic optical lattices,” J. Opt. A: Pure Appl. Opt. 9, S134–S138 (2007). [CrossRef]  

47. R. Dasgupta, S. Ahlawat, and P. Gupta, “Microfluidic sorting with a moving array of optical traps,” Appl. Opt. 51, 4377–4387 (2012). [CrossRef]   [PubMed]  

48. T. Čižmár, O. Brzobohatý, K. Dholakia, and P. Zemánek, “The holographic optical micro-manipulation system based on counter-propagating beams,” Laser Phys. Lett. 8, 50–56 (2011). [CrossRef]  

49. H. Risken, The Fokker-Planck Equation (Springer-Verlag, 1996).

50. P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002). [CrossRef]  

51. P. Zemánek, A. Jonáš, and M. Liška, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A 19, 1025–1034 (2002). [CrossRef]  

52. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004). [CrossRef]  

53. M. Šiler and P. Zemánek, “Particle jumps between optical traps in a one-dimensional optical lattice,” New. J. Phys. 12, 083001 (2010). [CrossRef]  

54. O. Brzobohatý, V. Karásek, M. Šiler, J. Trojek, and P. Zemánek, “Static and dynamic behavior of two optically bound microparticles in a standing wave,” Opt. Express 19, 19613–19626 (2011). [CrossRef]   [PubMed]  

55. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 2004). [CrossRef]  

56. H. A. Kramers, “Brownian motion in the field of force and the diffusion model of chemical reactions,” Physica 7, 284–304 (1940). [CrossRef]  

57. M. Šiler, T. Čižmár, A. Jonáš, and P. Zemánek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New. J. Phys. 10, 113010 (2008). [CrossRef]  

58. T. Imasaka, Y. Kawabata, T. Kaneta, and Y. Ishidzu, “Optical chromatography,” Anal. Chem. 67, 1763–1765 (1995). [CrossRef]  

59. C. C. Ho, A. Keller, J. A. Odell, and R. H. Ottewill, “Preparation of monodisperse ellipsoidal polystyrene particles,” Colloid Polym. Sci. 271, 469–479 (1993). [CrossRef]  

60. J. A. Champion, Y. K. Katare, and S. Mitragotri, “Making polymeric micro- and nanoparticles of complex shapes,” Proc. Natl. Acad. Sci. USA 104, 11901–11904 (2007). [CrossRef]   [PubMed]  

61. A. V. Arzola, P. Jákl, L. Chvátal, and P. Zemánek, “Rotation, oscillation and hydrodynamic synchronization of optically trapped oblate spheroidal microparticles,” Opt. Express 22, 16207–16221 (2014). [CrossRef]   [PubMed]  

62. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer-Verlag, 2011). [CrossRef]  

63. J. P. B. Schaub and D. R. Alexander, “Theoretical determination of netradiation force and torque for a spherical particle illuminated by a focusedlaser beam,” J. Appl. Phys. 66, 4594–4602(1989). [CrossRef]  

64. L. Tsang, Scattering of Electromagnetic Waves (Wiley, 2001).

65. F. Perrin, “Mouvement brownien d’un ellipsoide - i. dispersion dielectrique pour des molecules ellipsoidales,” J. Phys. Radium 5, 497–511 (1934). [CrossRef]