1. Introduction

Experiments on optical trapping using one single beam were only carried out in the middle 80’s, providing efficient full tridimensional trapping [1]. Since then, optical tweezers have found enormous applications in medical and biological researches [2–5].

It is widely expected that dielectric particles with refractive index n_p higher than that of the surrounding medium (n_m), be directed towards regions of high intensity of the impinging laser beam, due to the exerted gradient and scattering total forces. The contrary would happen if the particle had a refractive index lower than that of the medium. In this case, it would be repelled from these high intensity regions.

The explanation is based on momentum transfer from the incident photons to the particle, as illustrated in Fig. 1(a) . Considering the incident ray and only the first two refracted rays and using conservation of the momentum, it is quite straightforward to show that the particle will always be attracted into high intensity regions of the beam.

 

Fig. 1 (a). Ray optics analysis showing an incident ray R₁ and its first two refracted rays R₂ and R₃. Looking at the momentum transfer without considering the infinite series of refracted/reflected rays which appears as a consequence of R₁ leads to an erroneous interpretation and to the conclusion that the particle would experience a force directed to the high intensity region, i.e., to the centre of the beam. (b). the complete picture taking into account the infinite series of refracted/reflected rays.

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This truncated ray optics picture is, however, not completely correct. In fact, a sequence of infinite rays emerges from the particle after it is impinged by the incident ray, and the power of each of these rays is strongly dependent upon the relative refractive index n_rel between the particle and the surrounding medium (from Fresnel’s reflectivity and transmittivity coefficients), as can be appreciated in Fig. 1(b) for an incident ray with power P.

Although Ashkin confirms that scattering forces prevail over gradient forces for this situation [6] and subsequent works have found ways of trapping and manipulating these particles using two counter-propagating beams [7–10], here it is numerically demonstrated that even those clever apparatus [7–10] are unable to provide a fully tridimensional trap for particles with n_rel >> 1 because gradient forces can also become repulsive.

Let us start by analyzing a single incident ray to have an insight into the problem, then we will generalize the phenomenon for an incident Gaussian beam with wavelength λ much smaller than the dimensions of the particle, thus satisfying the ray optics requirements. Results are immediate and intuitive. A final example is provided for a Bessel beam, demonstrating that these multi-ringed shaped beams are well suited alternatives for trappings where n_rel >> 1.

The final conclusion is that optical trapping using one single beam and even counter-propagating beams for n_rel >> 1 are of limited use, and other experimental schemes or different kinds of laser beams may be necessary to be adopted.

2. Theoretical analysis – single ray incidence

Different from Fig. 1(a), the complete situation for an incident ray is depicted in Fig. 1(b), where the infinite series of reflected/refracted rays can be appreciated for n_p > n_m. Scattering and gradient forces (parallel and perpendicular to the ray, respectively) $F_S$ and $F_g$ are given by the known expressions [6]:

Using Fig. 1(a) for explaining the optical trapping of a particle is equivalent to take l’s upper limit as 0 in the above equations and to reject the first reflected ray given by the cos(π + 2θ_i) and sin(π + 2θ_i) terms for F_S and F_g, respectively, meaning that we are approximating the real situation.

Maybe one reason for using this approximation to explain the capabilities of optical tweezers in trapping biological particles is that most of these have refractive indices close to the surrounding medium. Thus, for a medium with n_m = 1.33, typical of experiments, R is satisfactorily low for highly focused beams with a numerical aperture less than 66°. But as n_rel increases, R increases as well, as can be appreciated in Fig. 2 . If R increases significantly, the first reflected ray carrying a power PR will dominate over all the other rays that emerge from the particle (multiple transmitted rays) with associate powers PT², PT²R and so on, all included in (1) and (2) by the summation over l. This happens because T = 1- R tends to zero.

 

Fig. 2 Reflectivity for n_m = 1.33 and n_p = 1.6, 2.4, 3.2 and 4.0 for (a) perpendicular polarization and (b) parallel polarization. If the laser is designed for case (b), then the incident rays can be chosen so that only the contribution of those with incidence angles close to regions of low reflectivity are relevant.

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Notice that, even if the scattering force could dominate over the gradient one for high refractive index, making optical trapping difficult to be achieved, here we find an inversion of the gradient force, that is, even if scattering forces could hypothetically be canceled, gradient forces become repulsive. Therefore, the statement that particles, having a refractive index higher than the medium, would be attracted to the optical axis of the beam is not always true, not only because of the well-known longitudinal pushing, but also because of this repulsive force perpendicular to the ray. So, Fig. 1(a) must be used cautiously when explaining the behavior of a particle in an optical tweezers system.

Still for a single ray, Fig. 3 shows the gradient force as a function of the incidence angle θ_i and n_p. It is clear that, according to the coordinate system in Fig. 1(b), when n_p increases, the gradient forces become positive in y, i.e., repulsive. Here, we suppose a circularly polarized ray, i.e., we use both R and T for perpendicular and parallel polarizations, taking the mean average in (2). It can be easily checked that, as n_rel (e.g., n_p for fixed n_m) → ∞, $Q_S\to 2{\cos }^2{\theta }_i$ and $Q_g\to \sin 2{\theta }_i$, i.e., there is a limit for the maximum force exerted by each ray and, consequently, for any designed beam or trapping setup in the ray optics regime. However, in this high n_rel case, all forces are purely repulsive.

 

Fig. 3 (a) Gradient Force (normalized over n_mP/c) for a single ray and its dependence on both incident angle and refractive index. (b) The equivalent contour plot, emphasizing the gradient zero-force line. As the refractive index increases, F_g becomes positive, i.e., repulsive, indicating the trapping impossibility.

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3. Theoretical analysis – beam incidence

Total forces for a Gaussian beam can be calculated as described elsewhere [6]. For a circularly polarized TEM₀₀ Gaussian beam with a numerical aperture of 66°, typical of the microscopes used in experimental setups, gradient and scattering total forces (considered here as the components of the total force vector, perpendicular and parallel to the optical axis of the beam, respectively) as functions of the distance r between the focus of the beam and the centre of the spherical particle for geometrical optics is shown in Fig. 4 for several values of n_p (n_m fixed). The optical axis of the beam is taken along the vertical + z coordinate with both focus and centre of the particle in a horizontal plane. The efficiency of optical trapping is significantly reduced as n_p increases. A particle of radius a = 10λ, λ = 1064 nm, is assumed.

 

Fig. 4 Gradient (a) and scattering (b) total forces for a circularly polarized TEM₀₀ Gaussian beam as functions of the distance r between the centre of the particle and the beam focus when both are on a horizontal plane perpendicular to the optical axis of the beam. The attractive/repulsive pattern depends on the value of the refractive index n_p for a fixed n_m.

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This does not means that high n_rel particles cannot be trapped; rather, it confirms the need for new experimental setups like the two counter propagating laser beams, or dual-beam trap, recently tested [9]. Notice, however, that even those arrangements are inadequate for the case of n_p = 6.4 in Fig. 4, because this circularly polarized beam would repeal the particle perpendicularly to the optical axis.

Referring to Fig. 5(a) , suppose that the impinging Gaussian beam propagates along -z and has a focus on the origin of the y-z plane. Any incident ray hits the spherical dielectric particle of vector radius a at some distance d from the focus with an incident angle θ_i. The angle between the z axis and the vector r which connects the focus and the center of the particle is denoted by γ. Total forces can be found by numerically summing up all individual gradient and scattering forces of each ray of the beam (see Ref [11]. for further details).

 

Fig. 5 (a). Coordinate system for total force numerical calculations. (b) Scattering factor Q_S and (c) gradient factor Q_g as functions of the angle γ between the optical axis of the beam and the vector connecting its focus to the centre of the particle. Higher repulsive scattering total forces are seen as n_p increases, whereas the gradient total forces become repulsive.

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A full analysis for a fixed distance between the center of the particle and the beam focus is provided based on the coordinate system of Fig. 5(a) for r = 0.5a and γ ranging from 0° to 360°. Results for scattering total forces are depicted in Fig. 5(b) for the same values of n_p as used before. These forces increase significantly over the equivalent gradient total forces in Fig. 5(c).

Notice that, for the chosen spherical particles and according to the convention of Fig. 5(a), the only trappable particle for a single beam would be the one with n_p = 1.6, as F_S becomes attractive (positive) towards the focus in the range 100° < γ < 260°, i.e., quite below the plane perpendicular to the optical axis and that crosses the focus.

There are, however, efficient ways of trapping high refractive index particles. Easy laser trapping could be achieved simply by replacing the counter propagating Gaussian beams by multi-ringed shaped beams, like Bessel beams. As the scattering forces are cancelled, gradient forces are enhanced, and several high refractive index particles could be efficiently trapped in the regions of low intensity profiles of these beams. Figure 6 illustrates the gradient total force profile for a particle with a = 10λ, λ = 1064 nm, as the distance r changes, for a zero-order Bessel beam with a spot of 28.89 μm, which is equivalent to an axicon angle of 0.0141 rad. In order to calculate the gradient total force, we again summed up all incident rays assuming that all of them are parallel to the optical axis of the beam and differing among them only in its carried power, P.

 

Fig. 6 Gradient total forces profile for a particle in a zero-order Bessel beam with λ = 1064 nm and a spot of 28.89 μm. The particle has a radius of a = 10.64 μm. For n_p = 8.0, points at r/a = 2.8 and 6.3 become points of stable equilibrium, and the use of two counter propagating Bessel beams present themselves as excellent alternatives for trapping these high refractive index particles.

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For this beam – and for a Bessel beam of any order – the radial repulsion from high intensity regions of the beam does not exclude trapping because the particle would tend to move to the low intensity zones, where it is eventually captured (in the case of Fig. 6, for example, at approximate distances 28 μm (r/a = 2,8) and 67 μm (r/a = 6.3) for n_p = 8.0, which are points of stable equilibrium).

It is difficult to find anything organic or biological having a refractive index higher than 3 [12], although at least one biological complex found in fungi presents an interesting behavior: when attempting to trap it using an optical tweezers, it is naturally repelled, even though its refractive index is higher than the surrounding medium, as confirmed by phase-contrast microscopy, where it appears as a phase-dark object [13,14]. This complex, called Spitzenkörper, seems to be important in fungi growth, but no refractive index has been assigned to it yet [15]. We may speculate that following [13,16,17] (λ ~785 nm and dimensions of the complex of the order of one micron), this complex could behave as an inhomogeneous particle having an overall effective refractive index much higher than expected, then being pushed away. Of course, other hypothesis may offer more plausible or effective explanations for this phenomenon.

All simulations in this paper were performed for spherical particles in the ray optics regime, and the limiting value between attractive/repulsive gradient forces can significantly change, for a given external medium, depending on the internal structure and the geometry of the particle and the ratio between its radius and the wavelength, i.e., the optical regime.

4. Conclusions

High refractive index particles request schemes of trapping different from the ones available in the literature. Even one single or two counter propagating Gaussian beams based on the strategy of scattering forces cancellation may not be capable of manipulating these particles because of the presence of repulsive gradient forces. Therefore, for n_rel >> 1, alternative experimental setups should be realized, as the use of two anti-parallel multi-ringed shaped Bessel beams, for instance.