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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Henry van Driel
  • Vol. 27, Iss. 8 — Aug. 1, 2010
  • pp: 1651–1658
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Geometrical optics calculation of forces and torques produced by a ringed beam on a prolate spheroid

Alberto Hinojosa-Alvarado and Julio C. Gutiérrez-Vega  »View Author Affiliations


JOSA B, Vol. 27, Issue 8, pp. 1651-1658 (2010)
http://dx.doi.org/10.1364/JOSAB.27.001651


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Abstract

The axial and transverse optical forces and torques exerted by circularly ringed beams on an arbitrarily oriented and homogeneous spheroid are calculated and studied within the framework of the geometrical optics regime. The results are applied to study the behavior of the forces in a counter-propagating optical trap. We calculate the trapping efficiencies and torques for several values of physical parameters, including the beam waist separation distance, the equivalent spheroid radius, the spheroid eccentricity, and the refractive index ratio between the particle and the surrounding medium.

© 2010 Optical Society of America

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

1. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975). [PubMed]

, 2

2. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–722 (1979). [CrossRef] [PubMed]

, 3

3. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995). [CrossRef] [PubMed]

, 4

4. J. P. Barton, “Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,” Appl. Opt. 40, 3598–3607 (2001). [CrossRef]

, 5

5. Y. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian beam illumination,” Appl. Opt. 42, 6621–6629 (2003). [CrossRef] [PubMed]

, 6

6. F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007). [CrossRef]

, 7

7. F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007). [CrossRef]

, 8

8. F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. E 78, 013843 (2008).

, 9

9. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40, 2501–2509 (2001). [CrossRef]

, 10

10. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430–443 (2007). [CrossRef]

, 11

11. R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008). [CrossRef]

, 12

12. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468–471 (2001). [CrossRef]

, 13

13. E. A. Hovenac, “Calculation of far-field scattering from nonspherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4746 (1991). [CrossRef] [PubMed]

, 14

14. F. Xu, K. F. Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. 45, 5000–5009 (2006). [CrossRef] [PubMed]

]. 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

1. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975). [PubMed]

, 2

2. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–722 (1979). [CrossRef] [PubMed]

, 3

3. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995). [CrossRef] [PubMed]

, 4

4. J. P. Barton, “Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,” Appl. Opt. 40, 3598–3607 (2001). [CrossRef]

, 5

5. Y. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian beam illumination,” Appl. Opt. 42, 6621–6629 (2003). [CrossRef] [PubMed]

, 6

6. F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007). [CrossRef]

, 7

7. F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007). [CrossRef]

, 8

8. F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. E 78, 013843 (2008).

], the scattering T-matrix theory [9

9. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40, 2501–2509 (2001). [CrossRef]

, 10

10. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430–443 (2007). [CrossRef]

, 11

11. R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008). [CrossRef]

, 12

12. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468–471 (2001). [CrossRef]

], and the geometrical optics framework [13

13. E. A. Hovenac, “Calculation of far-field scattering from nonspherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4746 (1991). [CrossRef] [PubMed]

, 14

14. F. Xu, K. F. Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. 45, 5000–5009 (2006). [CrossRef] [PubMed]

].

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 U(r)exp(iωt) propagates within a homogeneous medium with refractive index n0 along the z axis of a fixed laboratory reference frame r=xx̂+yŷ+zẑ. Consider a prolate spheroid with semi-major axis a, semi-minor axis b, and refractive index nsp, 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 r=dx̂+z0ẑ and introduce the center-of-mass (cm) frame rcm=xcmx̂cm+ycmŷcm+zcmẑcm with its origin 0cm 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 zsp the axis of revolution of the spheroid and by rsp=xspx̂sp+yspŷsp+zspẑsp the set of coordinates axes with origin at 0cm but fixed in the particle.

In the laboratory frame of reference, the position C=(Cx;Cy;Cz) of a point C on the surface of a prolate spheroid is written as
C=H(α,β)(bsinηcosφbsinηsinφacosη)+(d0z0),
(1)
where (η,φ) are the angle parameters of the prolate spheroidal coordinates, i.e., η[0,π] is the polar angle measured from the north pole of the spheroid, and φ[0,2π) is the azimuthal angle with respect to the axis zsp. The rotation matrix
H(α,β)=(cosαcosβsinαcosαsinβsinαcosβcosαsinαsinβsinβ0cosβ)
(2)
determines the orientation of the spheroid with respect to the laboratory frame; here α[0,2π) and β[0,π] are the azimuthal and polar angles of the spheroid axis zsp relative to the origin 0cm of the center-of-mass frame; see Fig. 1. Throughout the paper we consider a prolate spheroid, but we remark that the analysis can be straightforwardly extended to the case of oblate spheroids.

3. FORCES EXERTED BY A RINGED LIGHT BEAM

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 PT and the effective radius of the spheroid r0, namely
T=(n0PTr0c)τ=(n0PTr0c)jPjPTτ¯j,
(5)
where τ¯j is the normalized torque exerted by each ray, and r0 is the radius of a sphere with the same volume as the spheroid with semiaxes a and b, i.e., from the equality V=4πab23=4πr033 we get r0=(ab2)13.

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 PT, wavenumber k=n0ωc, and half-waist size w0 traveling along the propagation coordinate z:
LGm,l(r,θ,z)=2m!PTπ(m+l)!exp(ilθ)w(z)×[2rw(z)]lLml(2r2w(z)2)exp[r2w2(z)]×expi[kz+kr22R(z)(2m+l+1)ψGS(z)]x̂,
(6)
where Lml is the generalized Laguerre polynomial, the beam width reads as w2(z)=w02(1+z2zR2) with zR=kw022 being the Rayleigh range of the beam, ψGS(z)=arctan(zzR) is the Gouy shift, and R(z)=z(1+zR2z2) is the radius of curvature of the spherical wavefronts of the LG beam. The beam waist is located at z=0 in the coordinate system of the laboratory. It should be remarked that the scalar paraxial expression in Eq. (6) represents a linearly polarized component of the electric field emitted by an optical fiber; see, for example, [17

17. E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. 36, 6423–6433 (1997). [CrossRef]

].

At a given point r, the differential power of the LG beam flowing through a small region of the spheroidal surface is given by
Pj=I(rj)cosθjΔSj,
=2m!PTπ(m+l)!wj2(2rjwj)2l×[Lml(2rj2wj2)]2exp(2rj2wj2)cosθjΔSj,
(7)
where ΔSj is the differential surface on the illuminated part of the particle, and θj is the angle between the ray direction and the normal of the spheroid at the incident point.

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

18. M. Mansuripur, “Geometric-optical rays, Poynting’s vector and field momenta,” Opt. Photonics News 10(3), 53–56 (1999). [CrossRef]

]. The energy flux vector of a paraxial field U(r)=|U(r)|exp[iΦ(r)] points along the vector tΦ+kẑ, where 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

19. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000). [CrossRef]

]
ı̂=r2r̂+(lRk)θ̂+rRẑr4+l2R2k2+r2R2.
(8)

By substituting Eq. (7) into Eq. (4) the global trapping efficiency turns out to be
Q=jI(rj)cosθjPTqj,
=2m!π(m+l)!j(2rjwj)2lwj2×[Lml(2rj2wj2)]2exp(2rj2wj2)cosθjΔSjqj,
(9)
where qj is the trapping efficiency of the jth ray [15

15. H. Sosa-Martínez and J. C. Gutiérrez-Vega, “Optical forces on a Mie spheroidal particle arbitrarily oriented in a counterpropagating trap,” J. Opt. Soc. Am. B 26, 2109–2116 (2009). [CrossRef]

], Eq. (20). It is convenient to express the force and torque exerted by the beam in terms of the Cartesian components of the laboratory system shown in Fig. 1, that is,
Qx=Qx̂,Qy=Qŷ,Qz=Qẑ,
(10a)
τx=τx̂,τy=τŷ,τz=τẑ.
(10b)
(10)
To find the forces and torques, we need only to multiply these efficiencies by (n0PTc) and the torques by (n0PTr0c). Finally we remark, that in traditional analysis it is customary to average the effects of the TE and TM components of the ray at each reflection point. However, in our case, we take into account separately these effects in order to get more accurate results.

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=4πab23 is kept constant and equal to an equivalent sphere of volume V=4πr033, where r0 is the effective radius of the spheroid. The trapping efficiencies are calculated for various values of the eccentricity e=(a2b2)12a 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, Λ=(λ0n0)w0, is fixed at a value of Λ=0.04 (for example, this may correspond to a laser beam with w0=10μm and λ0=532nm and a spheroid suspended in water with n0=1.33). The relative refractive index and the effective radius of the particle are assumed to be nspn0=1.2 and r0w0=14, respectively. Note that the relation between the particle size and the wavelength in the surrounding medium is given by 2r0(λ0n0)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=z0w0=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=dw0. For comparison purposes, the actual size of the particle is also displayed in Fig. 2a. The transverse Qx and axial Qz 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 Qx=0. If the refractive index of the particle nsp is greater than the index of the surrounding medium n0, 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 Qx closely resembles the derivative of the intensity. The mode LG00 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., n0>nsp, the dark regions will become the stable equilibrium positions [20

20. V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002). [CrossRef]

, 21

21. K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002). [CrossRef]

, 22

22. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14, 4182–4187 (2006). [CrossRef] [PubMed]

].

On the other hand, the axial force Qz is shown in Fig. 2c. Qz is always positive for all values of D, and its maxima coincide with the maxima of the beam intensity. Note that Qz 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., LG00) over a spheroid with eccentricity e=0.6 located at the plane Z=z0w0=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 α=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., β=0,π) or orthogonal (i.e., π2) 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 β=π4 and β=3π4, the particle experiences a torque that changes its orientation to β=0 (or equivalently β=π) for which the torque vanishes. In this way, we conclude that when β=0,π is a situation of stable rotational equilibrium, whereas β=π2 corresponds to unstable equilibrium.

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

Figure 4 shows the trapping efficiencies and torques exerted by a LG10 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 β=0 and π2 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., β=0), but when the spheroid has a slight misalignment β>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 LG10 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 [α,β] of the spheroid. In the figure, the labels are normalized in units of π4, for example, the label [2, 1] means α=2π4 and β=π4. 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 LG01 beam for different values of β and α=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 LG10 beam.

Unlike the Gaussian and LG10 beams discussed before, the annular beam LG01 has an azimuthal dependence of the form exp(iθ); therefore it carries an orbital angular momentum of unity value (in units of ) in the direction of the z axis. As a consequence, the beam LG01 exerts an additional tangential force along the azimuthal direction θ̂, 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 [α,β] 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:
  • In general, for an initial arbitrary inclination, the major axis of a spheroid will tend to be aligned with the propagation axis of the beam with its centroid at a maximum of the intensity distribution.
  • If the major axis of the spheroid is not parallel to the z axis, the positions of stable equilibrium shift radially toward or away from the optical axis.
  • A spheroid in rotational equilibrium experiences weaker radial and longitudinal forces than a sphere with the same volume.
  • For beams with azimuthal dependence exp(ilθ), a spheroid at equilibrium is not aligned along the z axis, but it has a certain inclination in the direction of the ring. This inclination depends on the lateral displacement D.

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

17. E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. 36, 6423–6433 (1997). [CrossRef]

, 23

23. D. Rudd, C. López-Mariscal, M. Summers, A. Shahvisi, J. C. Gutiérrez-Vega, and D. McGloin, “Fiber based optical trapping of aerosols,” Opt. Express 16, 14550–14560 (2008). [CrossRef] [PubMed]

, 24

24. M. Guillon, O. Moine, and B. Stout, , and , “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); [CrossRef] [PubMed]

24. M. Guillon, O. Moine, and B. Stout, , and , erratum, Phys. Rev. 99, 079901 (2007).

, 25

25. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a-fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]

, 26

26. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: A novel laser tool to micromanipulate cells,” Biophys. J. 81, 767–784 (2001). [CrossRef] [PubMed]

, 27

27. G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976). [CrossRef]

]. 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 LG01 beams on a spheroid with eccentricity 0.6 for different orientations of the particle characterized by the angle β and α=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 LG01 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., 0.5w0). 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.
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=rw0. (b) (c) (d) Respectively, the Qr. Qz, and Qθ trapping efficiencies of a spherical particle immersed in the LG beams.
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 α=0.
Fig. 4 Trapping efficiencies and normalized torques exerted by a LG1,0 beam on (a) a spheroid of eccentricity 0.6 for different orientations β (b), (c), (d) of the spheroid (see Fig. 1) with α=0.
Fig. 5 Trapping efficiencies and normalized torques exerted by a LG1,0 beam on (a) a spheroid with different orientations [α,β] (b), (c), (d), (e) in units of π4.
Fig. 6 Trapping efficiencies and normalized torques exerted by a LG1,0 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 α=0.
Fig. 7 Trapping efficiencies and normalized torques exerted by a LG1,0 annular beam on (a) a spheroid of eccentricity 0.6 with different orientations [α,β] (b), (c), (d), (e) in units of π4.
Fig. 8 Trapping efficiencies and normalized torques exerted by two perfectly aligned counter-propagating beams LG1,0 on (a) a spheroid with different orientations β and α=0 (b), (c), (d), (e).
Fig. 9 Trapping efficiencies and normalized torques exerted by two misaligned counter-propagating beams LG1,0 on (a) a spheroid with different orientations (b), (c), (d) β and α=0. Subplots (e) and (f) show the cases of constructive and destructive superposition of the helical wavefronts, respectively.
1.

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975). [PubMed]

2.

S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–722 (1979). [CrossRef] [PubMed]

3.

J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995). [CrossRef] [PubMed]

4.

J. P. Barton, “Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,” Appl. Opt. 40, 3598–3607 (2001). [CrossRef]

5.

Y. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian beam illumination,” Appl. Opt. 42, 6621–6629 (2003). [CrossRef] [PubMed]

6.

F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007). [CrossRef]

7.

F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007). [CrossRef]

8.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. E 78, 013843 (2008).

9.

Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40, 2501–2509 (2001). [CrossRef]

10.

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430–443 (2007). [CrossRef]

11.

R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008). [CrossRef]

12.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468–471 (2001). [CrossRef]

13.

E. A. Hovenac, “Calculation of far-field scattering from nonspherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4746 (1991). [CrossRef] [PubMed]

14.

F. Xu, K. F. Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. 45, 5000–5009 (2006). [CrossRef] [PubMed]

15.

H. Sosa-Martínez and J. C. Gutiérrez-Vega, “Optical forces on a Mie spheroidal particle arbitrarily oriented in a counterpropagating trap,” J. Opt. Soc. Am. B 26, 2109–2116 (2009). [CrossRef]

16.

K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B 21, 1749–1757 (2004). [CrossRef]

17.

E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. 36, 6423–6433 (1997). [CrossRef]

18.

M. Mansuripur, “Geometric-optical rays, Poynting’s vector and field momenta,” Opt. Photonics News 10(3), 53–56 (1999). [CrossRef]

19.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000). [CrossRef]

20.

V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002). [CrossRef]

21.

K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002). [CrossRef]

22.

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14, 4182–4187 (2006). [CrossRef] [PubMed]

23.

D. Rudd, C. López-Mariscal, M. Summers, A. Shahvisi, J. C. Gutiérrez-Vega, and D. McGloin, “Fiber based optical trapping of aerosols,” Opt. Express 16, 14550–14560 (2008). [CrossRef] [PubMed]

24.

M. Guillon, O. Moine, and B. Stout, , and , “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); [CrossRef] [PubMed]

M. Guillon, O. Moine, and B. Stout, , and , erratum, Phys. Rev. 99, 079901 (2007).

25.

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a-fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]

26.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: A novel laser tool to micromanipulate cells,” Biophys. J. 81, 767–784 (2001). [CrossRef] [PubMed]

27.

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976). [CrossRef]

OCIS Codes
(140.7010) Lasers and laser optics : Laser trapping
(290.0290) Scattering : Scattering
(290.5850) Scattering : Scattering, particles

ToC Category:
Scattering

History
Original Manuscript: May 28, 2010
Revised Manuscript: June 23, 2010
Manuscript Accepted: June 28, 2010
Published: July 26, 2010

Virtual Issues
Vol. 5, Iss. 12 Virtual Journal for Biomedical Optics

Citation
Alberto Hinojosa-Alvarado and Julio C. Gutiérrez-Vega, "Geometrical optics calculation of forces and torques produced by a ringed beam on a prolate spheroid," J. Opt. Soc. Am. B 27, 1651-1658 (2010)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-27-8-1651


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References

  1. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975). [PubMed]
  2. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–722 (1979). [CrossRef] [PubMed]
  3. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995). [CrossRef] [PubMed]
  4. J. P. Barton, “Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,” Appl. Opt. 40, 3598–3607 (2001). [CrossRef]
  5. Y. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian beam illumination,” Appl. Opt. 42, 6621–6629 (2003). [CrossRef] [PubMed]
  6. F. Xu, K. Ren, G. Gouesbet, X. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E 75, 026613 (2007). [CrossRef]
  7. F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007). [CrossRef]
  8. F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. E 78, 013843 (2008).
  9. Y. Han, and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40, 2501–2509 (2001). [CrossRef]
  10. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430–443 (2007). [CrossRef]
  11. R. W. Going, B. L. Conover, and M. J. Escuti, “Electrostatic force and torque escription of generalized spheroidal particles in optical landscapes,” Proc. SPIE 7038, 703826 (2008). [CrossRef]
  12. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468–471 (2001). [CrossRef]
  13. E. A. Hovenac, “Calculation of far-field scattering from nonspherical particles using a geometrical optics approach,” Appl. Opt. 30, 4739–4746 (1991). [CrossRef] [PubMed]
  14. F. Xu, K. F. Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. 45, 5000–5009 (2006). [CrossRef] [PubMed]
  15. H. Sosa-Martínez, and J. C. Gutiérrez-Vega, “Optical forces on a Mie spheroidal particle arbitrarily oriented in a counterpropagating trap,” J. Opt. Soc. Am. B 26, 2109–2116 (2009). [CrossRef]
  16. K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B 21, 1749–1757 (2004). [CrossRef]
  17. E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. 36, 6423–6433 (1997). [CrossRef]
  18. M. Mansuripur, “Geometric-optical rays, Poynting’s vector and field momenta,” Opt. Photonics News 10(3), 53–56 (1999). [CrossRef]
  19. L. Allen, and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000). [CrossRef]
  20. V. Garcés-Chávez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002). [CrossRef]
  21. K. P. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4, S82–S789 (2002). [CrossRef]
  22. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14, 4182–4187 (2006). [CrossRef] [PubMed]
  23. D. Rudd, C. López-Mariscal, M. Summers, A. Shahvisi, J. C. Gutiérrez-Vega and D. McGloin, “Fiber based optical trapping of aerosols,” Opt. Express 16, 14550–14560 (2008). [CrossRef] [PubMed]
  24. M. Guillon, O. Moine, and B. Stout, “Longitudinal optical binding of high optical contrast microdroplets in air,” Phys. Rev. Lett. 96, 143902 (2006); M. Guillon, O. Moine, and B. Stout, erratum, Phys. Rev. 99, 079901 (2007). [CrossRef] [PubMed]
  25. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a-fiber-optical light-force trap,” Opt. Lett. 18, 1867–1869 (1993). [CrossRef] [PubMed]
  26. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: A novel laser tool to micromanipulate cells,” Biophys. J. 81, 767–784 (2001). [CrossRef] [PubMed]
  27. G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976). [CrossRef]

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