1. INTRODUCTION

In recent years, there has been an increasing interest in the theoretical treatment, measurement, and characterization of the optical forces exerted on particles with non-spherical shapes [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The relevance relies on the fact that the actual shape of real particles is not spherical. For instance, red blood cells, human sperm, and many other micro-organism cells cannot be adequately modeled by spheres. Additionally, shape distortion due to the softness of biological particles plays a significant role under the action of radiation pressure forces. A nonspherical model accounts better for this and other effects. One of the simplest models of the non-spherical particles is given by the spheroidal geometry. The scattering of light beams by spheroids has been studied by several authors using electromagnetic expansion methods [1, 2, 3, 4, 5, 6, 7, 8], the scattering T-matrix theory [9, 10, 11, 12], and the geometrical optics framework [13, 14].

In a previous paper [15], we calculated the axial and transverse optical forces exerted by Gaussian beams on an arbitrarily oriented spheroid within the framework of the geometrical optics regime. The results were applied to study the behavior of the forces in a counter-propagating optical trap. In this paper, we extend the analysis to calculate and discuss the optical forces and torques exerted by ringed light beams on a spheroid with arbitrary orientation. An interesting feature of ringed beams is that their alternate bright and dark regions allow the confinement of particles under different circumstances [16]. We particularly look at Laguerre–Gaussian beams, but stress that the results can also be applied to other ringed beams such as Bessel–Gauss beams or annular beams. Our findings allow us to make predictions concerning the radial equilibrium positions of the spheroidal particles as a function of their relative size with respect to the beam dimensions and the corresponding azimuthal forces.

2. DEFINITION OF THE PARAMETERS AND GEOMETRY OF THE PROBLEM

We briefly describe the geometry of the problem in order to establish notation and to provide a reference point for necessary formulas. An optical field $\mathbf{U}\left(\mathbf{r}\right)\exp (-i\omega t)$ propagates within a homogeneous medium with refractive index $n_0$ along the z axis of a fixed laboratory reference frame $\mathbf{r}=x\widehat{\mathbf{x}}+y\widehat{\mathbf{y}}+z\widehat{\mathbf{z}}$. Consider a prolate spheroid with semi-major axis a, semi-minor axis b, and refractive index $n_{\mathit{sp}}$, where a and b are assumed to be so much greater than the wavelength of the optical field that the ray-optics approximation can be applied.

We place the centroid of the spheroid at the arbitrary point $\mathbf{r}=-d\widehat{\mathbf{x}}+z_0\widehat{\mathbf{z}}$ and introduce the center-of-mass (cm) frame ${\mathbf{r}}_{\mathit{cm}}=x_{\mathit{cm}}{\widehat{\mathbf{x}}}_{\mathit{cm}}+y_{\mathit{cm}}{\widehat{\mathbf{y}}}_{\mathit{cm}}+z_{\mathit{cm}}{\widehat{\mathbf{z}}}_{\mathit{cm}}$ with its origin $0_{\mathit{cm}}$ fixed at the centroid of the spheroid, whose axes are parallel to the r frame as shown in Fig. 1 . The spheroid is obtained by rotation of an ellipse about an axis of symmetry. Let us denote by $z_{\mathit{sp}}$ the axis of revolution of the spheroid and by ${\mathbf{r}}_{\mathit{sp}}=x_{\mathit{sp}}{\widehat{\mathbf{x}}}_{\mathit{sp}}+y_{\mathit{sp}}{\widehat{\mathbf{y}}}_{\mathit{sp}}+z_{\mathit{sp}}{\widehat{\mathbf{z}}}_{\mathit{sp}}$ the set of coordinates axes with origin at $0_{\mathit{cm}}$ but fixed in the particle.

In the laboratory frame of reference, the position $\mathbf{C}=(C_x;C_y;C_z)$ of a point C on the surface of a prolate spheroid is written as

3. FORCES EXERTED BY A RINGED LIGHT BEAM

In the ray-optics regime the field $\mathbf{U}\left(\mathbf{r}\right)$ is treated as a bundle of individual rays representing infinitely localized conduits of power. The net force F acting on the spheroid is calculated by summing the contributions of the individual rays that are hitting the surface area of the particle; we have

Similarly, a net torque T acting about the center of mass of the spheroid is calculated by adding the individual torques produced by the rays that are hitting the surface of the particle. A normalized torque τ may be defined by normalizing the individual torques with respect to the total beam power $P^T$ and the effective radius of the spheroid $r_0$, namely

For the intensity distribution of the ringed light beam, we will consider a linearly polarized standard Laguerre–Gauss (LG) beam with radial m and angular l mode numbers, total power $P^T$, wavenumber $k=n_0\omega ∕c$, and half-waist size $w_0$ traveling along the propagation coordinate z:

At a given point r, the differential power of the LG beam flowing through a small region of the spheroidal surface is given by

Light rays can be defined as oriented curves whose direction coincides everywhere with the direction of the energy flux vector of the field, providing information about the magnitude and direction of the energy flow and the linear and angular momenta of the field [18]. The energy flux vector of a paraxial field $U\left(\mathbf{r}\right)=\left|U\left(\mathbf{r}\right)\right|\exp \left[i\Phi \left(\mathbf{r}\right)\right]$ points along the vector ${\nabla }_t\Phi +k\widehat{\mathbf{z}}$, where ${\nabla }_t$ stands for the transverse gradient. Replacing the corresponding values of a LG beam, the direction of the rays at the observation point r is given by the unit vector [19]

By substituting Eq. (7) into Eq. (4) the global trapping efficiency turns out to be

4. NUMERICAL CALCULATIONS

In this section, we present and discuss numerical results for the transverse and axial trapping efficiencies and normalized torques τ produced by ringed beams for a range of values of the normalized parameters. In order to make meaningful comparisons under different conditions, the semi-axes a and b of the spheroid may change, but its volume $V=4\pi a{b}^2∕3$ is kept constant and equal to an equivalent sphere of volume $V=4\pi r_0^3∕3$, where $r_0$ is the effective radius of the spheroid. The trapping efficiencies are calculated for various values of the eccentricity $e={(a^2-b^2)}^{1∕2}∕a$ of the spheroid and for different orientations of the particle characterized by the angles α and β shown in Fig. 1. For numerical purposes, the normalized wavelength in the external medium, $\Lambda =({\lambda }_0∕n_0)∕w_0$, is fixed at a value of $\Lambda =0.04$ (for example, this may correspond to a laser beam with $w_0=10\mu \mathrm{m}$ and ${\lambda }_0=532\mathrm{nm}$ and a spheroid suspended in water with $n_0=1.33$). The relative refractive index and the effective radius of the particle are assumed to be $n_{\mathit{sp}}∕n_0=1.2$ and $r_0∕w_0=1∕4$, respectively. Note that the relation between the particle size and the wavelength in the surrounding medium is given by $2r_0∕({\lambda }_0∕n_0)\approx 12.5$, which is enough to satisfy the requirement of the geometrical optics approximation.

4A. Forces on a Spherical Particle

To gain a physical insight, we first calculate the trapping efficiencies of a spherical particle located at the plane $Z=z_0∕w_0=10$. The radial variation of the intensity of several LG beams with different mode numbers $(m,l)$ is shown in Fig. 2a as a function of the relative transverse offset from the beam axis $D=d∕w_0$. For comparison purposes, the actual size of the particle is also displayed in Fig. 2a. The transverse $Q_x$ and axial $Q_z$ trapping efficiencies are depicted in Figs. 2b, 2c, respectively.

The radial forces define the equilibrium regions where the particles can be trapped radially within the beam intensity profile. In Fig. 2b, the equilibrium positions are located at positions where $Q_x=0$. If the refractive index of the particle $n_{sp}$ is greater than the index of the surrounding medium $n_0$, the equilibrium is stable when the slope of the curve is positive, which occurs about the intensity maxima of the beam, i.e., bright rings and the origin (for beams with $l=0$). The equilibrium is unstable when the slope of the curve is negative, i.e., dark rings and the origin (for beams with $l>0$). In general, the curve of $Q_x$ closely resembles the derivative of the intensity. The mode ${\mathrm{LG}}_{00}$ corresponds to a conventional Gaussian beam for which only at the origin can a particle be trapped. Conversely, if now the refractive index of the particle is lower than the refractive index of the surrounding medium, i.e., $n_0>n_{sp}$, the dark regions will become the stable equilibrium positions [20, 21, 22].

On the other hand, the axial force $Q_z$ is shown in Fig. 2c. $Q_z$ is always positive for all values of D, and its maxima coincide with the maxima of the beam intensity. Note that $Q_z$ never vanishes, even at the zero intensity points. Therefore, a single LG beam cannot be used to trap particles in three dimensions.

4B. Forces on a Spheroidal Particle: Gaussian Illumination

Figure 3 shows the variation in function of the lateral displacement D of the trapping efficiencies and torques exerted by a Gaussian beam (i.e., ${\mathrm{LG}}_{00}$) over a spheroid with eccentricity $e=0.6$ located at the plane $Z=z_0∕w_0=10$ for different orientations β; see Fig. 1. The intensity distribution of the light and the shape and size of the spheroid are depicted in subplot 3(a). The curves in subplots 3(b), (c), and (d) correspond to different orientations β of the spheroid with $\alpha =0$. In all cases, the dashed curves represent the behavior of a sphere with the same volume as the spheroid.

When the axis of the spheroid is parallel (i.e., $\beta =0,\pi$) or orthogonal (i.e., $\pi ∕2)$ to the z axis, the curves of efficiency and torque differ from the curve of a sphere only by a scale factor. However, for other inclinations of the spheroid, the curves are quite different with the consequence that the position of stable equilibrium moves away from the origin with a maximum radial excursion of about $D=0.5$.

The behavior of the normalized torque exerted by the beam about the center of mass of the particle is shown in subplot 3(d). This torque induces a rotation such that β increases. It can be observed for all values of D that when $\beta =\pi ∕4$ and $\beta =3\pi ∕4$, the particle experiences a torque that changes its orientation to $\beta =0$ (or equivalently $\beta =\pi$) for which the torque vanishes. In this way, we conclude that when $\beta =0,\pi$ is a situation of stable rotational equilibrium, whereas $\beta =\pi ∕2$ corresponds to unstable equilibrium.

4C. Forces on a Spheroidal Particle: Gaussian-Ringed Illumination

Figure 4 shows the trapping efficiencies and torques exerted by a ${\mathrm{LG}}_{10}$ beam. As shown in Fig. 4a, the intensity shape is formed by a central spot and a secondary bright circular ring. The particle now has two radial positions of stable equilibrium where it can be captured, i.e., the central maximum and the maximum of the ring. Like the case of a Gaussian beam, inclinations of the spheroid different from $\beta =0$ and $\pi ∕2$ produce a radial displacement of the equilibrium positions, e.g., about 0.2 and 0.1 for the central peak and the ring, respectively.

The curves in Fig. 4d for the torque reveal that, in this case, the rotational equilibrium does not occur when the major axis of the spheroid is aligned with the z axis (i.e., $\beta =0)$, but when the spheroid has a slight misalignment $\beta >0$, which depends on the lateral displacement D. For these particular values used in the simulation, the final stable inclination is negligible. At the central maximum, the transverse trapping force is stronger for the ${\mathrm{LG}}_{10}$ beam than for the Gaussian beam with the same waist size. Nevertheless, the trapping force at the first ring is almost an order of magnitude lower than the force at the central maximum, thus the beam has poor efficiency for trapping particles at the ring.

In Fig. 4 we have shown the curves of efficiency for different inclinations β of the spheroid. The inclination α (see Fig. 1) of the spheroid also affects the behavior of the trapping forces and torques. Figure 5 shows the behavior of the efficiencies and torques for different values of the orientations $[\alpha ,\beta ]$ of the spheroid. In the figure, the labels are normalized in units of $\pi ∕4$, for example, the label [2, 1] means $\alpha =2\pi ∕4$ and $\beta =\pi ∕4$. From the figure we can see that varying α leads to a scaling of the curves of efficiency, but the equilibrium positions do not move. The curves for the torque in Fig. 5d indicate that the particle will orient along the z axis independently of its initial orientation.

4D. Forces on a Spheroidal Particle: Ring-shaped with Orbital Angular Momentum Illumination

Figure 6 shows the trapping efficiencies and torques exerted by a ${\mathrm{LG}}_{01}$ beam for different values of β and $\alpha =0$. This case corresponds to a circular ring-shaped light beam with a null intensity center on the beam axis. There is only one radial equilibrium position at the maximum of the ring. The magnitudes of the forces are lower than for a Gaussian beam but higher than those for the bright ring of the ${\mathrm{LG}}_{10}$ beam.

Unlike the Gaussian and ${\mathrm{LG}}_{10}$ beams discussed before, the annular beam ${\mathrm{LG}}_{01}$ has an azimuthal dependence of the form $\exp \left(i\theta \right)$; therefore it carries an orbital angular momentum of unity value (in units of ℏ) in the direction of the z axis. As a consequence, the beam ${\mathrm{LG}}_{01}$ exerts an additional tangential force along the azimuthal direction $\widehat{\theta }$, which gives rise to the rotation of the particles around the beam axis.

The curves of the efficiencies and torques are depicted in Fig. 7 for several inclinations $[\alpha ,\beta ]$ of the spheroid. In this situation we can see that the steady state of the particle is not aligned along the z axis, but now it has a certain inclination in the direction of the ring.

4E. Physical Conclusions

From the situations discussed in the above sections we can extract the following main conclusions:

5. FORCES IN A DUAL-BEAM OPTICAL TRAP

Counter-propagating optical traps and their variants are motivated by the three-dimensional control, such as rotation of the particle, that can be exerted in a non-contact manner [17, 23, 24, 25, 26, 27]. In this section numerical results are presented for a dual-beam optical trap.

Figure 8 shows the trapping efficiencies and torques produced by two perfectly aligned counter-propagating ${\mathrm{LG}}_{01}$ beams on a spheroid with eccentricity 0.6 for different orientations of the particle characterized by the angle β and $\alpha =0$. The wavelength is assumed to be the same for both counter-propagating beams. The radial variation of the optical intensity at the observation planes is plotted in subplot 8(a). From the analysis we conclude that the longitudinal forces vanish only when the particle is located at the radial position of stable equilibrium.

For the case of perfectly aligned counter-propagating beams with non-zero orbital angular momentum (i.e., $l>0$) the sense of rotation of the helical wavefronts is of crucial importance. If the helical wavefronts rotate constructively, the azimuthal forces superpose and thus the particle tends to rotate around the optical axis z according to the direction of the rotation of the wavefronts. If the helical wavefronts rotate oppositely, the azimuthal forces tend to cancel out, and the particle does not experiment rotational force if it is located at the half-plane between both waist planes. If the spheroid is outside of this middle plane, then one of the beams will exert a stronger force than the other, and there will exist a tendency to rotate. Finally, in contrast to the case of a single beam, for the case of perfectly aligned counter-propagating beams, the radial positions of equilibrium are invariant under a change of the orientation β of the spheroid, and are the same as those for the equivalent sphere.

The trapping efficiencies and torques produced by two parallel misaligned counter-propagating ${\mathrm{LG}}_{01}$ beams are depicted in Fig. 9 . Subplot 9(a) shows the intensities of the beams at the observation plane. The optical axes of the beams are parallel with an offset given by a half of the beam width at the waist size (i.e., $0.5w_0$). The first consequence of the misalignment is that the equilibrium point moves to the middle point between the intensity maxima of the beams. The major axis of the spheroid experiences a tendency to be aligned along the propagation axis z. If the helicities of the helical wavefronts superpose constructively, the particle still tends to rotate in this direction, but the azimuthal force is no longer constant along a transverse circle of fixed radius with center at the optical axis. In general, the azimuthal forces are of one order of magnitude lower than the longitudinal and radial forces.

6. CONCLUSIONS

A study of the optical forces exerted by ringed beams on a spheroidal particle has been presented. The size of the particle was assumed to be so much larger than the light wavelength that the ray-optics approximation could be employed. Optical forces and torques on a spheroidal particle were also analyzed for the case of a dual ringed-beam trap. In this paper we focused our attention on beams with axisymmetric structure. However, the analysis can be straightforwardly extended to optical beams with different transverse structure.

ACKNOWLEDGMENTS

We acknowledge support from Tecnológico de Monterrey (grant CAT141) and from Consejo Nacional de Ciencia y Tecnología (grant 82407).

 

Fig. 1 Parameter definitions and geometry of the problem.

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Fig. 2 (a) Radial variation of the intensity of several LG beams with different mode numbers $(m,l)$ as a function of the normalized radius $D=r∕w_0$. (b) (c) (d) Respectively, the $Q_r$. $Q_z$, and $Q_{\theta }$ trapping efficiencies of a spherical particle immersed in the LG beams.

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Fig. 3 Trapping efficiencies and normalized torques exerted by a Gaussian beam on (a) a spheroid of eccentricity 0.6. The curves in subplots (b), (c) and (d) correspond to different orientations β of the spheroid (see Fig. 1) with $\alpha =0$.

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Fig. 4 Trapping efficiencies and normalized torques exerted by a ${\mathrm{LG}}_{1,0}$ beam on (a) a spheroid of eccentricity 0.6 for different orientations β (b), (c), (d) of the spheroid (see Fig. 1) with $\alpha =0$.

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Fig. 5 Trapping efficiencies and normalized torques exerted by a ${\mathrm{LG}}_{1,0}$ beam on (a) a spheroid with different orientations $[\alpha ,\beta ]$ (b), (c), (d), (e) in units of $\pi ∕4$.

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Fig. 6 Trapping efficiencies and normalized torques exerted by a ${\mathrm{LG}}_{1,0}$ annular beam on (a) a spheroid of eccentricity 0.6 for different orientations β (b), (c), (d), (e) of the spheroid (see Fig. 1) with $\alpha =0$.

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Fig. 7 Trapping efficiencies and normalized torques exerted by a ${\mathrm{LG}}_{1,0}$ annular beam on (a) a spheroid of eccentricity 0.6 with different orientations $[\alpha ,\beta ]$ (b), (c), (d), (e) in units of $\pi ∕4$.

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Fig. 8 Trapping efficiencies and normalized torques exerted by two perfectly aligned counter-propagating beams ${\mathrm{LG}}_{1,0}$ on (a) a spheroid with different orientations β and $\alpha =0$ (b), (c), (d), (e).

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Fig. 9 Trapping efficiencies and normalized torques exerted by two misaligned counter-propagating beams ${\mathrm{LG}}_{1,0}$ on (a) a spheroid with different orientations (b), (c), (d) β and $\alpha =0$. Subplots (e) and (f) show the cases of constructive and destructive superposition of the helical wavefronts, respectively.

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