1. Introduction

Optical binding [1–3] denotes an important type of light-matter interaction between mesoscopic particles that is mediated by the light re-scattered by the particles. Regardless of the terminology of use (i.e. ”binding”) this interaction leads to attractive or repulsive forces between the particles and, therefore, can lead to their self-arrangement into so called “optical matter” [4]. This intriguing self-arrangement is based upon delicate equilibrium between the optical forces resulting both from the incident beam and from the light re-scattered by the particles that results in well-defined stable equilibrium positions for the particles concerned. From the technical point of view this mechanism is different from the concept of optical trapping because here the particles create stable structures due to the interaction between the particles and not due to the predefined shaping of the trapping beam defining an equilibrium position (optical trap) for each particle. The mere presence of the objects in the field modifies the field spatial distribution so that possible stable positions and configurations are created for interacting particles. The inter-particle separations are usually comparable to the wavelength of the illuminating laser beam and, therefore, mainly long-range interactions mediated by the far-field scattered field are considered while the near-field coupling is usually omitted in the related studies. The aim of the field of optical binding is to understand these phenomena in depth and to pave the way for controlled self-assembly of colloidal particles solely by light radiation.

Two distinct cases of optical binding have been investigated in the previous studies. So-called lateral binding corresponds to the configuration where the binding beam impinges perpendicularly to the inter-particle axis [1,4–11]. On the contrary longitudinal binding geometry uses two counter-propagating incoherent beams and the optically bound particles are self-arranged along the axis of beams propagations while kept at the beam center by lateral intensity gradient of the beam [12–30]. A special instance of longitudinal binding is particle self-arrangement near a surface under the illumination of counter-propagating evanescent waves [31–38].

In this paper we present a novel geometry where, due to the presence of reflective surface in the sample chamber, both the lateral and the longitudinal optical binding effects are simultaneously contributing to the process of particle self-organization. Moreover, altering the angle of the beam incidence we can modify the conditions for stable particle self-organization.

A deeper quantitative understanding of these effect would allow one to engineer appropriate conditions for self-assembling of structures at will or modify properties of already assembled soft matter structures. Our study includes both numerical and experimental approaches to understand these phenomena but due to very high complexity of these problems we address only the cases of two and three optically bound objects. The paper is organized as follows. Section 2 presents the optical binding fields in more details and the following section 3 introduces our numerical approach based on the enhanced coupled dipole method. Section 4 introduces the experimental procedure we developed for measurements of optical binding forces and the comparison between numerical and experimental data.

2. The optical binding geometry

Figure 1 introduced the geometry of the optical field and the experimental set-up. The optical binding effects were established under the influence of a wide Gaussian beam at the wavelength of 532 nm incident onto the sample under a variable angle. This beam was reflected at the bottom surface (dielectric mirror). Both the incident and the reflected beams interfered and created a standing wave along the axis perpendicular to the bottom mirror. The optical binding effects were studied in proximity of the top transparent surface assuring a constant distance of the formed structures from the bottom reflective surface.

 

Fig. 1 A suspension of polystyrene particles of diameter 820 nm was placed between the microscope slide (top) and the cover-glass (bottom). The cover-glass was coated with a system of dielectric layers (SiO₂ and TiO₂) reflecting 99% of the incoming beam. This incoming wide Gaussian beam responsible for optical binding (green color, vacuum wavelength 532 nm, beam waist radius 20μm placed at the mirror, incident power 600 mW, Coherent Verdi V5). The interference of the incoming and reflected beam creates a standing wave along y axis over the distance of 20μm between the sample boundaries (set by polystyrene spheres of diameter 20μm). The incident angle θ of the binding beam was controlled in the range of 0–4 degrees by the movable mirror. The second laser pathway at 1064 nm (red color, power in the sample 50–150 mW, IPG ILM-10-1064-LP) acts as a time-shared multiple optical trapping system used for precise force measurement of optical binding forces. The optical trap is formed by oil immersion objective (Nikon Plan Apo, oil, NA 1.4, 100x) and positioned by an acousto-optic deflector (AOD IntraAction DTD-276HD6M driven by two synchronized NI PCI - 5412 RF signal generators). A Basler GigE CCD camera Basler piA640*210gm is used to record the position of the particles. The following focal lengths of the lenses are used in the paths: f₁ = 100 mm, f_2–4 = 300 mm, condenser lens 25 mm, tube lens in the microscope pathway 200 mm.

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Crucially, the employment of the reflective surface represents a principal difference from the standard lateral geometry. Since any particle illuminated by the incident beam modifies the light by scattering and absorption, this feature directly affects all particles in the vicinity of the scatterer (lateral binding), but crucially the excitation is reflected on the surface together with the incident beam and thus affects the scattering particle itself as well as other particles from other directions (longitudinal binding) as shown in the Fig. 2. Altering the angle of the beam incidence (see Fig. 1) we can redirect the excitation and redistribute the resulting field of optical forces and thereby control the conditions for particle self-organization.

 

Fig. 2 Detailed interaction between the dipoles via the scattered light only and the geometry of the configuration with two particles considered. Left: All single-pathway optical interactions between two dipoles A and B – In addition to the single direct interaction between dipoles (A⇌B) three other pathways are considered due to the reflection of the scattered light at the mirror. Middle: Consideration of the mirror images of the dipoles to include the influence of the mirror for the scattered light. Right: Geometry of the configuration with two particles – Green lines denote direction of the incident and the reflected binding beams placed in the y – z plane with the angle of incidence θ. The inter-particle axis is aligned along the z axis and the centre of mass C of the bound particles is placed at (x,y,z) = (0, y_C, z_C). y_C and z_C are the coordinates of the centre of mass C measured from the point where the the centre of the incident beam reflects at the mirror surface.

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The quantitative measurements of optical binding forces were attained by employment of a secondary multiple optical trapping system at a wavelength of 1064 nm. The procedure together with the experimental data is introduced in the Section 4.

3. Numerical modelling of optical binding effects

We developed a numerical model based upon the coupled dipole method (CDM) [6, 19, 39–41] that was enhanced by the fast Fourier transform method [42] and existing symmetries in the configuration. This method is based upon dividing each particle into elementary, mutually interacting dipoles placed on an orthogonal grid with inter-dipole distances so small that the field in the vicinity of each dipole can be considered uniform. Based on the criterion that the inter-dipole distance should be smaller than 1/10 of the wavelength inside a particle [43] we use the distance between the dipoles equal to 25 nm because no significant difference of the results was observed for shorter inter-dipole distances. Using this method the interaction of the electromagnetic field with microobjects is not restricted to homogeneous isotropic spheres but arbitrarily shaped particles can be modelled [44]. The CDM calculates the distribution of dipole moments of the individual dipoles forming the particles and the total force acting upon the particle is obtained as the sum of the forces acting upon these dipoles [45]. The binding force between two selected particles is equal to the difference between the total forces acting upon each particle keeping the convention that negative (positive) binding force pushes the particles closer to (further from) each other.

In this particular configuration the field scattered by the particles is also reflected at the mirror. Therefore, the optical binding force is composed of two contributions, the first one is given by the scattered field with no reflection at the mirror and the second contribution has its origin in the field scattered by one dipole and reflected at the mirror towards the second dipole. This second scattering pathway was included into the CDM code using the mirror images of the dipoles created by the mirror. This approximation is justified because the dielectric mirror behaves, for all practical purposes, like an ideal mirror with negligible absorption, zero surface modes, and further the dipoles are placed relatively far from the surface of the mirror. This procedure was implemented to address any interaction between dipoles, i.e. including dipoles in single particle and in different particles, too. For clarity these contributions are illustrated in Fig. 2 as the interaction between two optically interacting dipoles A and B. For example the field radiated by the dipole A towards dipole B through the reflection at the mirror is the same as the field at the place of the mirror image dipole B’ if the appropriate Fresnel reflection coefficients are included. This enhancement of the code enabled us to investigate the influence of the mirror reflections on the optical binding between a number of particles. The positions of the particles and related geometrical parameters of interest are defined in the right part of Fig. 2.

The theoretical results for the binding force quoted in this article are expressed as a function of the inter-particle separation z_sep because the same dependence was studied experimentally as a function of several parameters of the system. Figure 3 presents the dependence of the optical binding force between two particles on the inter-particle separation z_sep for s and p polarizations of the incident binding beam. In the case of s-polarization, p-polarization the electric field is oriented perpendicularly ⊥, parallel || to the connecting line between the particles, respectively. The same figures also present the strong influence of the mirror on the optical binding between both particles. Fast oscillations with decreasing amplitude correspond to the direct lateral binding between particles without the influence of the mirror and their period corresponds to the wavelength λ of the binding laser in water - as it is expected for lateral binding [1, 2]. Amplitude of these oscillations is higher for s-polarized incident beam because the induced dipoles in both small particles are oriented such that they scatter the wave strongly towards the other particle. This gives stronger optical interaction between them and consequently stronger binding force [2]. This type of interaction occurs between particles illuminated by the incident beam.

 

Fig. 3 Dependence of the optical binding force F_bind on the inter-particle separation z_sep for s (left) and p (right) polarization of the incident beam. The dashed and the solid curves denote the results with the interaction through the mirror omitted or included, respectively (see Fig. 2). Parameters of the calculations are: polystyrene particles (with 820 nm in diameter and refractive index of 1.59) placed 20 μm above the mirror (y_C = 20μm), angle of incidence θ = 0°, power of the laser beam in the colloidal suspension is 600 mW, vacuum laser wavelength is 532 nm, beam waist of the incident Gaussian beam (w₀ = 20μm) is placed at the mirror, refractive index of the water medium is 1.33, and the reflectivity of the mirror is 99%.

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The mirror reflects the light scattered by the particles in the forward direction. For the considered particle size the intensity of the forward scattered light is several orders of magnitude higher comparing to the laterally scattered light. Therefore, the scattered waves reflected at the mirror cause stronger binding interaction with amplitudes overcoming even the original lateral optical binding force magnitudes and introduce slow modulation of the binding force profiles.

Figure 4 shows the influence of the vertical placement of the bound particles (y_C, see Fig. 2) on the binding forces. Since the slow oscillations with noticeable peaks are caused by the scattered waves reflected at the mirror, analogously with the right geometrical view in Fig. 2 they move towards larger z_sep with increasing y_C and, consequently, their magnitude decreases. The dashed red curve presents the binding force profile for comparison if the mirror influence is neglected.

 

Fig. 4 Influence of the distance from the mirror y_C on the binding force profile F_bind between two particles. Peaks of slow oscillations move towards larger z_sep with increasing y_C and their magnitude decreases. Without the presence of the mirror (thick dashed red curve) two particles are predominantly attracted (negative sign of the force). In the case of s polarization the amplitude is again much stronger and several stable separations of two particles can be established. The parameters of the calculation are the same as in Fig. 3 except the incident angle θ = 2°.

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Figure 5 illustrates how the binding force profile is influenced if the particles are not placed symmetrically in between the incident and the reflected beams (z_C ≠ 0, see Fig. 2) for perpendicularly (θ = 0°) and tilted (θ = 2°) incident beam. In the case of perpendicular illumination (θ = 0°) the shift of the binding force does not depend on the sign of z_C but only on its magnitude. This effect is quite subtle and is caused by the Gaussian radial profile of the incident beam which slightly increases particles repulsion. The following first-order explanation can be adopted for particles displaced much less that the Gaussian beam waist. If both particles are deviated from the central position (z_C ≠ 0), the particle placed further from the beam centre feels stronger radial gradient of the optical intensity causing its stronger attraction towards the beam centre (for particles much smaller than the trapping wavelength it is denoted as the gradient force). Vice versa, the other particle feels weaker radial gradient of the optical intensity and weaker attraction towards the beam centre. In the studied case the difference between these forces contributes to the inter-particle binding force which leads to weaker particles attraction comparing to the initial symmetric configuration (z_C = 0). Therefore, the final binding force dependence is shifted towards positive values as Fig. 5 demonstrates.

 

Fig. 5 Influence of the lateral placement z_C of the bound structure with respect to the incident beam on the binding force profile F_bind between two particles. The thick red line corresponds to a symmetric configuration of the particles (z_C = 0) with respect to the incident and reflected beams. The parameters of the calculation are the same as in Fig. 3 except for the incident angle that is set to θ = 0° and 2°, respectively. The vertical placement of the bound structure is fixed at y_C = 18μm.

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For the selected incident angle θ = 2° the axes of the incident and reflected beams are separated only by 1.3 μm at the distance y_C = 18μm from the mirror. Therefore, for the purposes of the following explanation both overlapping beams could be considered as a single beam with Gaussian radial profile of the optical intensity and the explanation presented above can be used. However, nonzero incident angle induces nonzero component of the wavevectors of incident and reflected beams along the inter-particle axis (i.e. z-axis). Consequently, the nonzero component of the optical force appears which is associated with the radiation pressure along z-axis (for particles much smaller than the trapping wavelength it is denoted as the scattering force). This force component, acting upon each particle, is always positive for considered θ > 0° but its magnitude depends on the position of the particle because it is proportional to the value of the optical intensity at the position of the particle. Again the difference between both such forces creates the contribution to the binding force, however, this contribution is negative for z_C > 0 and positive for z_C < 0 (see Fig. 5).

We conclude this theoretical study with Fig. 6 which presents the stable equilibrium inter-particle positions (i.e. the positions where the binding force is zero and has negative slope with respect to z_sep) as the function of the angle θ of the incident beam. It is seen that this parameter influences the distance between two bound particles and can be used for experimental tuning of the optically bound one-dimensional colloidal structures. Note that the stability of such structures (with respect to a perturbation from the presented stable equilibrium positions) increases if higher power of the incoming laser is used. Obviously, more stable equilibrium positions (stable structures) can be found for s-polarized incident beams due to stronger direct interaction between the particles (lateral binding). This polarization selection provides stable structures with particles placed closer to each other (smaller z_stab) compared to the p-polarized case. This figure also indicates that p-polarized beams would be advantageous for practical realization of colloidal structures with better-defined inter-particle distances which can be more easily controlled by altering the incident angle θ.

 

Fig. 6 Stable equilibrium positions of two optically bound particles for different incident angles θ. The circle marks ○ denote the most stable equilibrium positions done by the intersection of the slow modulation with zero-force line and the cross marks × denote all the other equilibrium positions caused by the direct lateral binding between the particles for s-polarized incident beam. All the other parameters of the calculation are the same as in Fig. 3.

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4. Experimental measurement of optical binding forces

For quantitative measurements of the optical binding forces we introduced an optical trapping pathway controlled by an acousto-optic deflector to confine a number of particles in a timesharing manner [46]. In our case this flexible tool was used to arrange a number of particles placed near the top surface with different inter-particle distances. The optical traps suffered from strong spherical aberration due to large axial depths in the sample that precluded any axial trapping (along the y direction) with the used oil immersion objective. This was, however, desirable for our needs as we needed the particles to be in a close proximity to the top surface of the sample chamber. The magnitude of the trapping forces in the lateral direction (x – z) was set several times stronger that the optical binding effects to assure the particles do not escape from the optical traps. The optical stiffness (the spring constant) was, however, small enough to analyse the optical binding forces. The dependence of optical trapping force on the inter-particle separation was obtained as follows: To calibrate the stiffness of the trapping system for each measured configuration we arranged the particles into this structure solely by the trapping system while the binding optical field (laser at 532 nm) was turned off. From the recorded Brownian motion of the trapped particles we determined the stiffness of each optical trap κ_i using the equipartition theorem [47]: κ_i = k_BT/〈z²〉_i, where k_B denotes Boltzmann constant, T absolute temperature, and 〈z²〉_i the mean square displacement of the particle in i-th trap. Subsequently, the binding beam was turned on and the new mean values of particles’ positions were determined (see Fig. 7). The deviation of j-th particle from its equilibrium position due to the presence of the binding beam ${〈z〉}_j^{\text{on}}-{〈z〉}_j^{\text{off}}$ revealed the external optical force acting upon each particle due to the presence of the binding force: $F_j=-{\kappa }_j\left({〈z〉}_j^{\text{on}}-{〈z〉}_j^{\text{off}}\right)$. The binding force acting between individual particles j and k can be determined: F_bind = F_j – F_k. Negative (positive) binding force means the particles are pushed closer to (further from) each other.

 

Fig. 7 Time record of the deviations of j-th particle from its equilibrium position in the cluster of two optically bound particles if the binding laser is turned off ( ${〈z〉}_j^{\text{off}}$, red) or on ( ${〈z〉}_j^{\text{on}}$, green). Each part contains 1000 positions and the transition of the particles to their new positions took about 250 ms and these data were omitted from further processing.

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The above described procedure was repeated for two and three particles arranged in a line where the inter-particle distances were varied from 1.8 μm to 8 μm in 100 nm steps. Each experiment was performed for both polarizations of the binding beam and for incident angles θ varying between 0° – 4°. The power of the laser beam at the region of colloidal suspension was equal to 600 mW. The experimental results are plotted in Figs. 8 and 9 for two and three polystyrene spheres aligned in a line, respectively.

 

Fig. 8 Comparison of measured (blue curves) and calculated (red curves) binding forces F_bind between two polystyrene particles for s (left column) and p-polarized (right column) incident beam. In the case of s-polarization (mode ⊥: the electric field is polarized perpendicularly to the connecting line between the particles) the binding forces exert strong oscillations with period equal to λ. These effects are suppressed in the || mode because the radiation between the particles is weaker for the p-polarization. The forces were measured for incident angles θ_exp in the range 0–4 degrees. For the theoretical prediction the assumed distance from the mirror y_C and the incident angle θ_cdm were chosen within the calculated grid to give the best coincidence with the measured values. The remaining parameters of the calculation are the same as in Fig. 3. Note, that due to a malfunction of our motorized stage during the measuring of binding force – p - polarization, θ = 0° – we show for this case data from different experimental series. In this series the separation between the particles varied between (1.8 – 5.5) μm.

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Fig. 9 Comparison of binding forces between the left and the middle particle (1 – 2) and the middle and the right particle (2 – 3). In the case of s-polarization (mode ⊥: the electric field is polarized perpendicularly to the connecting line between the particles) the binding forces exert strong oscillations with period equal to λ. These effects are suppressed in the || mode because the radiation between the particles is weaker for the p-polarization. The accordance between the measured values (blue lines) and calculated ones (red lines) is getting worse for larger angles of incidence θ.

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4.1. Optical binding between two spheres

The experimental and theoretical evaluation of binding forces between two interacting spheres are presented in Fig. 8. In this comparison we used just two free parameters to get the presented coincidence. As we demonstrated in Fig. 4 variation of the distance y_C between the particles and the mirror by 50% significantly influences the shape and amplitude of the binding force profiles. Unfortunately, we have not been able to measure precisely y_C during the experiment and, therefore, we varied this parameter in 2 μm steps in the theoretical model and looked for the best coincidence with the experimental results. Similarly based on Fig. 6 the changes of the angle of incidence in 0.5° steps causes shift of the zeros of the binding force up to tens percent. The angle of incidence in the experiment θ_exp was set by the shift of the movable mirror with expected experimental error up to 0.3°. Therefore we changed the angle of incidence θ_cdm, entering the theoretical model, in 0.5° steps. Varying both free theoretical parameters, y_C and θ_cdm we looked for such combination that gave the best coincidence between the theoretical and experimental results. All the other theoretical parameters (i.e. laser power, beam waist, particle sizes, refractive indices of the particles and water) were the same as in Fig. 3 and they corresponded to the measured experimental values or suppliers’ data sheets.

Let us briefly discuss their possible influence on the comparison. One can see from Fig. 5 that the shift of the particles from the symmetrical placement with respect to the beams could contribute to the final binding force because the position of the particles with respect to the beam centre was measured with precision of about 3 μm. The amplitude of the binding force is directly proportional to the incident laser power but, of course, the equilibrium positions are not influenced by this parameter. The laser power (600 mW) was measured in the air at the plane of the particles position and the uncertainty of its measurement is less than 3%. This value cannot significantly modify the presented comparisons. The beam waist of the incident beam (20μm) was quantified using the fit of the Gaussian intensity profile on the calibrated CCD image of the beam. We estimate uncertainty of its determination less than 5%. Even though we have not studied theoretically the influence of the beam waist variation on the binding force profile, we do not expect that such tiny deviations in the beam width could significantly influence the comparison if all the considered length scales (inter-particle distances, particle sizes) are significantly smaller that the beam waist. We consider the values of the refractive index and size of the particles as the most probable sources of the discrepancies between the theory and experiments. The supplier’s data sheet provides particle diameter with standard deviation 6% (we have not verified this value experimentally). In order to take this deviation into account we have calculated the binding forces between a pair of particles up/down - sized by 6%, respectively. This tiny size variation leads to approximately 35% increase/decrease of the binding force amplitude. Similarly, it is known that the variation of the refractive index of the particle can significantly modify the binding force profile [19], however, we have not been able to measure the refractive index of the particle with sufficient precision. Our calculations have shown that the increase/decrease of the refractive index of the particles by 1% leads to approximately 15% increase/decrease of the binding force amplitude. However, both variations of the refractive index and the size of the particles, have not significantly shifted the particles equilibrium positions. Heating of the particles and the surrounding medium, during the measurement, alters and can cause disagreement in the presented comparisons.

We found the overall coincidence between the theoretical and experimental results fairly good considering the complexity of the problem and magnitudes of the binding force with respect to the total optical force acting upon each particle. Especially, the oscillations caused by the direct lateral binding are clearly visible here.

4.2. Optical binding between three spheres

Figure 9 presents the results of the first quantitative measurement of the binding forces between three particles and their comparison with the theoretical model. We considered the same distance y_C of all three particles from the mirror and symmetrical arrangement of the particles with respect to the beams i.e. z_C = 0. We used the same procedure as above and we looked for the best coincidence between the experimental and theoretical results if the distance y_C between the particles and the mirror was altered. The rest of the parameters were fixed including θ = θ_cdm = θ_exp. The first sight reveals that the profiles are different for the left-hand and right-hand pairs of optically bound particles for non-zero incident angles. This is caused by asymmetrical scattering between the particles for non-zero incident angles. The coincidence between the theory and experiment is very good even for p-polarized configuration, however, as in the case of two spheres gets worse with increasing angle of beam incidence. Nevertheless, we found the coincidence between the theory and the experiment satisfactory and even though there are scaling and offsetting deviations, the theoretical and the experimental data reveal the same behaviour of optical binding and particle self-organization.

5. Conclusion

We have presented a complex quantitative analysis of a novel experimental geometry with strongly enhanced optical binding effects due to the presence of a reflective surface on the sample boundary. We observed surprisingly strong optically mediated interactions between scattering objects that could be exploited to establish stable optical bonds between particles with a tuneability given by the flexible angle of the incident binding beam. We developed a new experimental protocol that allows determination of the optical binding forces acting upon microparticles. This protocol could be also used for mapping of internal forces of any origin, such as hydrodynamical coupling forces or screened Coulombic interactions, in the colloidal systems that are out of equilibrium. The experiment was performed in a controlled way that allowed parametric study of the problem, namely we changed the distances between the particles, the angle of incidence and polarization of the binding beam. The experimentally determined binding forces were compared with the theoretical predictions of an enhanced coupled dipole method. These results represent the first detailed and well-controlled measurement of optical binding forces and at the same time their comparison with theoretical predictions. Both the theoretical and the experimental data revealed the possibility of multi-stable optical binding in this geometry with considerably larger magnitudes in comparison to the case of pure lateral binding. We believe that this approach represents an important step towards controllable formation of optically bound matter as well as enhancement and deeper understanding of optical binding possibilities.