1. Introduction

Sub-micron colloidal particles can be trapped at an interface by the evanescent wave of a totally internally reflected laser beam. With a single laser beam [1], there is a tangential component to the radiation pressure that drives the colloidal particles along the surface. If two counter-propagating laser beams are employed, the spatially averaged radiation pressure is zero and stable trapping is observed [2, 3, 4]. For polystyrene spheres (PS) in water trapped at the silica-water interface, these particles assemble into lines parallel to the plane of incidence (for diameters 2a>700 nm) and into two-dimensional arrays for smaller particles (300<2a<700 nm). In a preliminary communication [3] we reported the formation of either hexagonal or rectangular arrays depending on the particle size and the presence or absence of interference between the two laser beams. In this paper, we explore in greater detail the effect of the polarization of the laser beams on the structure of 2-D arrays. We show the competition between optical trapping and optical binding [5] leads to some new and unexpected structures.

2. Experimental methods

A 500-mW, CW, SLM diode-pumped Nd:YVO4 laser (Laser Quantum, Forte-S, λ=1064 nm) was gently focused into a fused silica prism at an angle just above the critical angle (θ_c=67°) for total internal reflection at the silica–water interface (see Figs. 1 and 2). A half-wave plate was used to control the polarization of the incident beam, either s or p. After exiting the back face of the prism, the laser beam was retro-reflected by a spherical gold mirror that refocuses the beam to the same point on the silica surface as the incident beam. The spot size was ca. 15×30 µm at the surface where optical trapping occurs. A quarter-wave plate was used to select the polarization of the reflected beam, either parallel or perpendicular to the incident beam. With the quarter-wave plate set to rotate the polarization by 90°, less than 1% of the reflected beam was passed by a polarizer set parallel to the incident beam: the two laser beams are therefore accurately orthogonal. A calculation of the Fresnel coefficients for reflection at the silica-water interface shows that the y-component, E_y, of the electric field (for s-polarized light) and the z-component, E_z, of the electric field (for p-polarized light) were equal in magnitude to within the accuracy to which the angle of incidence was set. At angles close to the critical angle, E_x << E_z for p-polarized light.

 

Fig. 1. Schematic of optical path for the trapping of particle arrays in an evanescent field.

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Fig. 2. Close-up illustration of the prism and thin film used for trapping. The quarter-wave plate is used to control the relative polarization of the incident and reflected beams.

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A dilute suspension of PS spheres (Agar Scientific and Bangs Laboratories) was placed on the surface of the prism and covered with a microscope coverslip. The PS spheres were illuminated from below with white light, imaged through a ×100 oil-immersion objective and recorded with a video camera.

3. Experimental Results

3.1 S-polarized light

The decay length, d_p, of the evanescent field (independent of polarization) into the aqueous medium is of the order of the particle size (d_p ~800 nm). There is therefore a strong electric field gradient that acts to confine the PS particles to a plane parallel to the silica-water interface. Since both the silica surface (at pH=7) and PS spheres carry a negative surface charge, there is a double-layer repulsion at sufficiently short distance that prevents the particles from sticking to the surface.

When both the incident and reflected beams are s-polarized, the two fields interfere and create a set of interference fringes perpendicular to the plane of incidence (the y-direction). The periodicity of these fringes in the x-direction is ~396 nm (the wavelength of the laser in water is 800 nm and the lasers are incident just above the critical angle). For particles with a diameter substantially less than the wavelength of the light [6] there is a strong gradient force that acts to trap the particles on the interference fringes. The expectation is therefore for the formation of rows of particles parallel to the interference fringes [7]. What we observed is the formation of lines perpendicular to the fringes, which grew into two-dimensional chess-board arrays as more particles diffused into the evanescent field (Fig. 3).

 

Fig. 3. Array of 460-nm diameter spheres with s-polarized light. A centred rectangular unit cell is shown, with lattice parameters a and b perpendicular and parallel to the fringes, respectively

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The periodicity of the arrays in the x-direction was equal to twice the fringe spacing (~792 nm), while the periodicity in the y-direction decreased as the diameter of the particles decreased. The lattice dimensions are summarized in Table 1 for four particle sizes in 5 µM NaCl.

Table 1. Mean lattice parameters for three particle sizes in 5 µM NaCl. Errors in a and b are ±5 nm.

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The 520 nm particles were purchased from a different supplier (Agar) than the other sizes (Bangs Labs) and appear to bear a lower surface charge, resulting in a smaller separation between the particle surfaces at equilibrium. Increasing the ionic strength (which screens the electrostatic repulsions between the charged latex spheres) decreased the particle separation parallel to the fringes (b lattice parameter) but did not affect the separation perpendicular to the fringes (a lattice parameter). This demonstrates that the minimum in the optical binding potential lies at a smaller particle separation than the equilibrium value, which is where the optical binding and electrical double layer forces are balanced. The chess-board motif is common to all the particle sizes that form extended 2-D arrays (2a=600–300 nm). In the case of the smallest particles studied (2a=300 nm) the compression of the unit cell in the b direction is such that the packing is approaching hexagonal, but the lack of contrast in the optical image and the large amount of Brownian motion precluded the measurement of the lattice parameters.

A remarkable observation is that under identical conditions to those under which the chess-board arrays are formed (Fig. 3), the 520 and 460 nm particles may also form an array with distorted hexagonal symmetry that is incommensurate with the interference fringes. An example is shown in Fig. 4 of 520 nm particles in 5 µM NaCl. We label this packing ‘hex 1’. The unit cell parameter perpendicular to the fringes in Fig. 4 is 567±5 nm, compared to a fringe spacing of 396 nm. The chess-board array was usually formed initially and would then occasionally transform to the hex 1 array. Transformations from ‘hex 1’ to square were not observed. These observations suggest that there are multiple minima in the potential surface, with the square packing being the kinetic product but the ‘hex 1’ array being the thermodynamic minimum. The hex 1 array was not observed for 390 nm particles.

 

Fig. 4. Hexagonal array formed by 520 nm particles in s-polarized light with 5 µm NaCl

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3.2 P-polarized light

Under p-polarized illumination, two unreported structures were observed, again co-existing under the same experimental conditions. The first of these, illustrated in Fig. 5(a) for 520 nm particles, is a hexagonal array, ‘hex 2’, rotated by 30° compared to the hex 1 array in Fig. 4. This hex 2 array is commensurate with the fringes with every second fringe occupied by particles. The nearest neighbor spacing, 895 nm, is significantly greater than in the square arrays formed under s-polarization (610 nm): the hex 2 array is not close-packed. The hex 2 structure readily converted to a ‘broken’ hexagonal structure in which every third fringe is unoccupied (Fig. 5(b)). The b parameter contracted from 816 nm in the hex 2 array to 778 nm in the broken hex 2 array, though we have not measured a large enough number of images to establish whether the difference in b parameters between the two arrays is significant. In the broken hex 2 structure the particles in the two filled columns are separated by greater than the fringe spacing; probably with one displaced to the right of the fringe and the other to the left. These broken arrays were quite unstable with columns of particles readily jumping into the vacant fringes, in a concerted move affecting a number of column pairs.

Rotation of a half-wave plate in the incident laser beam converts the polarization smoothly from s to p and back to s again. The arrays were observed to transform from square packing to broken hex 2 and back to square as the waveplate was rotated.

 

Fig. 5. Arrays formed by 520 nm particles in p-polarized light: (A) hex 2 structure in which every second fringe is occupied; (B) broken hex 2 structure in which every third fringe is unoccupied. Spacing of interference fringes is shown by red lines.

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Neither the hex 2 nor the broken hex 2 structure was observed with 420 nm and 390 nm particles, which formed square arrays with the similar lattice parameters in both s and p-polarized light (Fig. 6).

 

Fig. 6. Square array formed by 390-nm diameter particles in p-polarized light.

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3.3 Orthogonal polarization

If a quarter-wave plate set at an angle of ϕ=45° to the fast axis is placed between the prism and the retro-reflector, the s-polarized incident light is converted to p-polarized by two passes through the quarter-wave plate. The polarizations of the two evanescent fields are then orthogonal and there are no interference fringes: the only spatial variation in the electric field arises from the broad Gaussian envelope of the laser beams. Under these conditions both 520 and 460 nm particles were observed to form distorted hexagonal arrays with the hex 2 orientation. The hex 2 array under orthogonal polarization is much more closely packed than that formed under p-polarization (Fig. 5(a)) and the lattice parameters are not simple multiples of the wavelength of the light, as might be expected from a dipole model of optical binding [12].

 

Fig. 7. Pseudohexagonal array formed by 460 nm particles in orthogonally polarized laser beams.

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A phase transformation from the square to hex 2 structure can be induced by rotation of the quarter-wave plate (Fig. 8). The sequence of video frames shown in Figure 8 shows the hexagonal packing nucleating at the centre of the array (Fig. 8(b)) and then growing outwards toward the left and right-hand sides of the array. A histogram was constructed of the angle of the waveplate, ϕ, at which the phase transition occurred. The peak of the distribution was at ϕ=30° for 520 nm particle and ϕ=35° for 460 nm particles. There was no hysteresis in this angle, depending on the direction of the phase transformation.

 

Fig. 8. Sequence of video frames of an array of 520 nm particles as a quarter-wave plate is rotated to change the polarization of the retro-reflected beam from s (ϕ=0°) to p (ϕ=45°)

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In contrast to the larger particles, the 390 nm particles adopted the same square-packing in orthogonally polarized light as in s and p-polarization. The a lattice parameter was still equal to twice the fringe spacing, even in the absence of fringes. The presence or absence of fringes did, however, have a marked effect on the dynamics of the arrays. In s-polarized light, individual PS spheres would diffuse up and down the fringes and only occasionally hop from one fringe to another. The result was that the arrays often contained defects, such as vacancies, missing rows and twin planes, that were slow to heal (see Fig. 9(a)). In orthogonally polarized beams, individual particles could diffuse freely in the surface plane and defects in the arrays rapidly healed (Fig. 9(b)).

 

Fig. 9. Arrays of 390 nm particles with (A) and without (B) interference fringes, showing the different types of defects that arise in the presence of interference fringes. a–missing row, b–vacancy, c–twin plane. The vacancies in (B) healed after a few seconds by diffusion to the edge of the array.

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4. Discussion

The formation of two-dimensional arrays at the silica–water interface can be considered in terms of five forces, four of which have their origin in scattering of the incident laser radiation.

1. The evanescent wave penetrates a distance ~λ into the aqueous medium. This rapidly decaying electric field gives rise to a gradient force that attracts the polystyrene spheres towards the silica-water interface. Numerous authors have calculated the force on a dielectric sphere in an evanescent field [8–11]. For practical purposes, the evanescent nature of the interfacial field serves to constrain the particles to lie in a plane parallel to the surface.

2. Both the PS spheres and the silica surface bear a negative charge at neutral pH. There is therefore a repulsion between particles due to the electrical double layer. The range of this repulsion can be tuned by variation in the electrolyte concentration. The experiments here were conducted in NaCl with concentrations of 5 µM to 1 mM, for which the corresponding Debye lengths are 150–10 nm. For the dilute salt solutions, this value of the Debye length is an overestimate, since it does not allow for the existence of ions present in the commercial colloidal suspension, for ions that leach from the coverslip or for hydrogen carbonate arising from dissolved CO₂. The double layer repulsion prevents the particles from sticking to each other or to the silica surface. Variation in the electrolyte concentration also changes the effective size of the particles (see below).

3. The Gaussian laser beam is weakly focused at the interface to a spot with dimensions of ~15×30 µm. The beam profile generates a gradient force that helps to draw particles towards the centre of the laser focus. We do not believe that this gradient force is important in array formation for several reasons. First, isolated particles were observed to diffuse relatively freely within the laser beams (in one or two dimensions according to the presence or absence of interference fringes). Second, the symmetry and orientation of the arrays has a strong dependence on polarization and particle size, whereas the gradient force from the envelope of the lasers alone would be expected to give hexagonal packing of random orientation. Third, multiple arrays readily coexist in the laser focus (Fig. 10), whereas the gradient force would collapse the particles into a single array.

 

Fig. 10. Three coexisting arrays of 460 nm particles in s-polarized light.

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4. When the incident and reflected beam have the same polarization, interference fringes are generated perpendicular to the plane of incidence with a fringe spacing of around 390 nm. The gradients in intensity associated with these fringes are two orders of magnitude larger than those arising from the Gaussian envelope of the laser beams. Particles experience a sinusoidal potential from the fringes, with the minimum in energy coinciding either with the fringe maxima or fringe minima, depending on the value of ka, (where k=2πn/λ and n is the refractive index of the water). Lekner [6] found that the particles with ka<2.85 (2a=725 nm in our experiments) the minimum in the potential occurs when the particles are located on the fringe maxima [6]. Ng and Chan [16] find that the change-over from fringe maxima to minima (and hence the point at which the particles experience no tangential gradient force from the interference fringes in the evanescent field) occurred at 2a=500 nm. We have not established experimentally whether the particles sit on fringe maxima or fringe minima, since the objective is not achromatic over the wavelength interval from the visible to the 1064 nm. Optical trapping of particles by the interference fringes is most clearly observed in the one-dimensional diffusion of isolated particles along the fringes and by the slow rate of hopping from fringe to fringe.

5. Optical binding refers to the interaction between particles induced by the optical field, in contrast to optical trapping which refers to the interaction of a particle with the applied optical field. In the Rayleigh limit (ka<<1), optical binding can be thought of as arising from the interaction between the induced dipoles on the particles [12, 4]. More generally, in the Mie scattering regime (which applies to the sizes of spheres described in this paper) it arises from multiple scattering of the incident field by the spheres. Binding between two Mie particles (ka=1) in an evanescent wave has been treated by Chaumet and Nieto-Vesperinas [4] for the case where the particles lie in the plane of incidence (separation in the x-direction). They found oscillations in the interaction potential between the two particles with a separation of ~λ/4 between the minima. These minima were at the same separations for s and p-polarized light.

The problem of calculating self-consistently the interaction potential between large numbers of Mie scatterers in an electromagnetic field is formidable. Several groups are beginning to make progress on this problem [13–15]. It is hoped that the observations reported here will provide a stimulus to theoreticians to model our experimental geometry. Nevertheless, we can make a number of general observations on the influence of optical trapping and optical binding in our experiments.

The simplest case to address is that of s-polarized light in which all particles with 300 nm≤2a≤520 nm form rectangular lattices that are commensurate with the interference fringes, with a lattice parameter equal to twice the fringe spacing. The particles are constrained by optical trapping to lie on the interference fringes. Decreasing the Debye length decreases the particle spacing, from which we infer that the interparticle force due to optical binding is attractive. For 390 nm particles at low ionic strength (5 µM NaCl) (Table 1) or 460 nm particles at high ionic strength (1 mM NaCl) the lattice is almost square (Fig. 11(b)), with a nearest neighbor spacing between centers of 560 nm. As the effective size of the particle is increased, either by increasing the actual particle size or by lowering the ionic strength, the lattice expands in the direction parallel to the fringes. When b=√3a, the lattice is hexagonal (Fig. 11(a)). This limit is approached by particles with 2a=620 nm, though extended 2D-arrays of these larger particles were difficult to form. Conversely, decreasing the effective particle size leads to compression of the b lattice parameter, such that the lattice for 300-nm particles is almost hexagonal again (b=a/√3). The relative orientations of the primitive unit cells in Figs. 11(a) and 11(c) are rotated by 30°: the hex 1 structure has the orientation of Fig. 11(a), while the hex 2 structures have the orientation of Fig. 11(c).

 

Fig. 11. Commensurate lattices formed in s-polarized fields. B is a square lattice. A is a hexagonal lattice formed by expansion of unit cell in b direction. C is a hexagonal lattice formed by contraction of unit cell in b direction. Centered rectangular unit cell shown in red; primitive hexagonal unit cells in blue dashes.

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That optical trapping and optical binding can be of comparable strengths is illustrated by the formation of the competitive hex 1 structure (Fig. 4), which is incommensurate with the interference fringes. The commensurate structure was generally formed first, but on occasions converted to the incommensurate hex 1 structure. The reverse transformation was not observed spontaneously, suggesting that the hex 1 structure was of lower energy: optical binding won out over optical trapping. The calculations of Ng and Chan suggest a reason why the hex 1 structure was observed for 460 and 520-nm particles, but not for 420 or 390 nm particles [16]. They find that amplitude of the spatially oscillating trapping potential is smaller for the two larger particle sizes, since they are close to the diameter (~500 nm) where the minimum of the potential switches from fringe maxima to fringe minima and hence where the corrugation in the trapping potential disappears.

In p-polarized light, only commensurate structures were observed. The 520-nm particles spontaneously formed a hexagonal structure, similar to Fig. 11(c) but with twice the unit cell in each direction: the particles are too large to sit on every fringe. The broken hex 2 structure can be understood as a compromise between optical binding, favoring a close-packed hexagonal structure and optical trapping, favoring locations on the fringe maxima. The smaller 460-nm particles preferentially adopted the more densely packed broken hex 2 structure. The two smallest particle sizes that could be clearly resolved (2a=420 and 390 nm) did not exhibit a hexagonal structure, but preserve the rectangular packing observed with s-polarized light. This behavior may be ascribed to the stronger corrugation potential experienced by these smaller particles.

Finally, under orthogonally polarized light, the 520 and 460-nm diameter particles formed a regular close-packed structure that was close to perfect hexagonal symmetry. The transformation from hexagonal to square packing upon rotation of the quarter-wave plate occured more readily for 460 than 520-nm particles, consistent with calculations showing a stronger corrugation with the 460-nm particles in s-polarized light [16].

The observation of hexagonal packing for particle sizes with the weakest corrugation in the trapping potential might suggest that optical binding in these 2D arrays favors hexagonal symmetry, with the orientation of the array dependent on the polarization of the electric field. It is therefore puzzling that the 390-nm particles retain a square geometry even in the absence of interference fringes. One might ascribe this behavior to weak residual interference fringes in orthogonally polarized beams, but this explanation seems unlikely: the polarization of reflected laser beam was accurately perpendicular to the incident beam and the beams were only weakly focused (f/#=8) so extensive mixing of polarizations due to focusing did not occur.

5. Summary

We have shown that two-dimensional arrays with a variety of different structures can self-assemble in the evanescent field of two counter-propagating laser beams. The symmetries and lattice parameters of the arrays depends on the particle size and polarization of the laser beams. Phase transitions between arrays of different symmetry can be induced by rotation of the polarization. The observed behavior can be rationalized qualitatively in terms of competition between optical trapping, leading to structures commensurate with the interference fringes, and optical binding due to multiple scattering of the incident fields by the particles. A quantitative explanation of the experimental observations will require detailed calculations of the optical binding potentials for arrays of different spacing and symmetry.