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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 24287–24292
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Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime

Leonardo A. Ambrosio and Hugo E. Hernández-Figueroa  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 24287-24292 (2010)
http://dx.doi.org/10.1364/OE.18.024287


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Abstract

Gradient forces on double negative (DNG) spherical dielectric particles are theoretically evaluated for v-th Bessel beams supposing geo-metrical optics approximations based on momentum transfer. For the first time in the literature, comparisons between these forces for double positive (DPS) and DNG particles are reported. We conclude that, contrary to the conventional case of positive refractive index, the gradient forces acting on a DNG particle may not reverse sign when the relative refractive index n goes from |n| > 1 to |n| < 1, thus revealing new and interesting trapping properties.

© 2010 OSA

1. Introduction

During the past two decades, optical tweezers have been extensively under investigation and, as far as ordinary particles are concerned – i.e., particles with positive refractive index -, its theory is fully established and plenty of applications have been proposed and successfully experimented, especially in biomedical optics, including, for example, biological cells manipulation such as trapping of viruses and bacteria [1

1. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]

], induced cell fusion [2

2. R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991). [CrossRef] [PubMed]

], studies of chromosome movement [3

3. M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86(12), 4539–4543 (1989). [CrossRef] [PubMed]

] and cellular microscopy [4

4. V. Emiliani, D. Cojoc, E. Ferrari, V. Garbin, C. Durieux, M. Coppey-Moisan, and E. Di Fabrizio, “Wave front engineering for microscopy of living cells,” Opt. Express 13(5), 1395–1405 (2005). [CrossRef] [PubMed]

].

The principle behind optical trapping with one single beam is usually explained by considering the momentum transfer from the photons to an isotropic, linear and homogeneous spherical particle and, in the ray optics regime, total gradient and scattering forces can be easily calculated by summing up the force contributions of all individual rays impinging the particle [5

5. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992). [CrossRef] [PubMed]

]. Alternatively, electromagnetic theory can be used as a general rule for calculating the optical forces for any optical regime using the Mie scattering and the beam shape coefficients, the latter being associated with the arbitrary incident beam [6

6. G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7(6), 998–1007 (1990). [CrossRef]

].

But what would happen if the particle were of double negative (DNG) nature, i.e, np or, equivalently, n, were negative? In a recent paper [9

9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009). [CrossRef] [PubMed]

], gradient forces where calculated for a single incident ray in the ray optics regime, revealing new and interesting properties. It was shown, for example, that they present both repulsive and attractive values, depending on the incidence angle, regardless of any particular fixed |n|, and a repulsive pattern for incident angles above some critical angle (see, e.g., Fig. 2
Fig. 2 A v = 0 Bessel beam (solid line) and Fg for several values of n (np). The surrounding media has nm = 1.33. Note that, as expected, in trapping DPS spherical particles, Fg inverts sign when n goes from n < 1 to n > 1.
of ref [9

9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009). [CrossRef] [PubMed]

].). From this, first results were obtained for a DNG spherical particle trapped by a focused Gaussian beam, being further extended to other regimes by adopting the so-called generalized Lorenz-Mie theory together with the integral localized approximation [10

10. L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express (to be published). [PubMed]

].

In this paper we calculate, for the first time in the literature, the total gradient forces exerted on a (simple, lossless and non-magnetic) DNG spherical particle by a v-th order Bessel beam Jv(.), using geometrical optics. Multi-ringed beams offers several advantages over focused beams, such as the simultaneous trapping of several biological particles of arbitrary shapes and sizes [11

11. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001). [CrossRef]

,12

12. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]

] including trapping in multiple planes [13

13. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]

]. Predicting how a DNG spherical particle would be trapped by these beams amounts to an increasing knowledge of what could be called “double-negative optical trapping”.

2. Theoretical analysis

Figure 1
Fig. 1 Coordinate system for trapping a DNG spherical particle using a v-th order Bessel beam.
shows the coordinate system considered in our simulations. The optical axis of the beam is parallel to + z, meaning that all rays impinging the spherical particle comes from –z. In the ray optics regime, we can assign a specific power P to each of these rays according to a Bessel beam profile, as follows:
P(r,θ,ϕ)=|Jv(kρr2sin2θ+ρ022rρ0sinθcos(ϕϕ0))|2
(1)
(r,θ,ϕ) being spherical coordinates relative to the center of the particle, kρ the radial wavenumber of the beam and ρ 0 and ϕ 0 defined according to Fig. 1.

The gradient force exerted on a DNG spherical particle by a single ray is
Fg=nmP(r,θ,ϕ)c{Rsin2θiT2[sin(2θi+2θt)+Rsin2θi]1+R2+2Rcos2θt}
(2)
where R and T are the Fresnel coefficients of reflection and transmission, respectively, θi the incident angle of the incident ray and θt the angle of the transmitted ray [9

9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009). [CrossRef] [PubMed]

]. The difference between (2) and the gradient force in the DPS case lies solely on the positive sign in the argument 2θi + 2θt, because all infinite reflected/transmitted rays follow the inverted Snell’s law, due to np being negative [14

14. V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]

]. Total gradient forces are found by numerically integrating (2) over the impinged hemisphere of the DNG particle, in a process analogous to that of a focused Gaussian beam [9

9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009). [CrossRef] [PubMed]

]. A right-hand circularly polarized beam is assumed.

Suppose a Bessel beam with a wavelength λ = 1064 nm, an axicon angle α = 0.0141 rad or, equivalently, a spot of Δρ = 28.89 μm and a radial wavenumber kρ = 83261.3 m−1. A dielectric, homogeneous, isotropic and linear spherical particle with radius a = 10λ is assumed throughout in all simulations. These parameters were chosen in order to ensure the ray optics requirements and that the scalar power formula approximation, given by (1), is completely within its range of validity.

For these parameters, Fig. 2 shows the total gradient force profile expected when the particle is DPS for ϕ 0 = −π (Fg = Fx), v = 0, nm = 1.33 and several values of n = np/nm. Because of ϕ 0, negative Fx means an attractive force. The intensity profile of the beam is also plotted as a solid line. Note the common repulsive/attractive pattern according to the intensity of the beam and its reversion when n = np/nm goes from n > 1 to 0 < n < 1. For n > 1, points of stable equilibrium occurs at ρ 0 ≈0, 46, 84 μm and all subsequent high intensity regions of the beam (not seen in the picture), whereas for 0 < n < 1, those points are ρ 0 ≈30, 67 μm and so forth. It is obvious that Fx would be zero for all ρ 0 whenever n = 1 (matched case).

Due to the inversion of Snell’s law, however, total gradient force presents an unusual trapping profile, as can be appreciated in Fig. 3
Fig. 3 Same results as in Fig. 2 for a DNG particle. For all five relative refractive indices depicted, the points of stable equilibrium are about the same, located at distances close to nulls of intensity (again, the intensity profile is shown as a solid line).
for the same parameters of Fig. 2. Note that, because Fx ≠ 0 for n = −1, this curve is also presented for completeness. In Figs. 4
Fig. 4 A v = 3 Bessel beam (solid line) and Fx for four values of n. Again, nm = 1.33. Negative values of Fx means a gradient force pulling the particle towards the optical axis of the beam. For a DPS particle, the points of stable equilibrium depends upon n being higher or less (but higher than zero) than one.
and 5
Fig. 5 A v = 3 Bessel beam (solid line) and Fx for four values of n. Again, nm = 1.33. For this DNG particle, points of stable equilibrium are seen at ρ 0 ≈0, 76, 117 μm regardless of |n| being higher or less than one.
, results for Fx over a DPS and a DNG particle are shown for v = 3, respectively. In all cases, positive values of Fx means that this force is repulsive, and vice-versa. To ensure that our theoretical proposal is adequate, all simulations were compared with those obtained by means of the generalized Lorenz-Mie theory (GLMT) with the integral localized approximation [15

15. L. A. Ambrosio, and H. E. Hernández-Figueroa are preparing a manuscript to be called “Integral localized approximation description of ordinary Bessel beams in the generalized Lorenz-Mie theory and application to optical forces.”

,16

16. L. A. Ambrosio, and H. E. Hernández-Figueroa, “Integral localized approximation description of v-th order Bessel beams in the generalized Lorenz-Mie theory and applications to optical trapping,” in Proceedings of PIERS2011in Marrakesh (to be published).

]. Good agreement was achieved.

To explain these results in view of the geometrical optics, one must look at the graphical representation of (2) for both |n| < 1 and |n| ≥ 1 and compare it with the DPS case, as we have done before for a focused Gaussian beam (Fig. 2 of ref [9

9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009). [CrossRef] [PubMed]

].). Depending on the incident angle of the rays that composes the beam, the gradient force exerted on the particle by each ray can be either attractive or repulsive. Consider the optical axis of the Bessel beam in Fig. 1 as before and, furthermore, the DNG particle at an arbitrary ρ 0 < Δρ (although we have assumed ϕ 0 = −π, due to symmetry and the optical regime adopted, the choice of ϕ 0 is irrelevant). If the overall presence of rays with an attractive nature relative to this axis, i.e., towards the optical axis, overcomes the repulsive effect (away from the optical axis), then the DNG particle will be directed towards lower values of ρ 0. The same qualitative analysis is valid for ρ 0 > Δρ.

This is exactly what is seen in Fig. 3 for ρ 0 < Δρ and all subsequent regions where the total gradient force is positive: for a zero-order Bessel beam and the parameters chosen, the DNG particle will always be directed towards low intensity regions of the beam (nulls of intensity) and there it will remain trapped, regardless of |n| being higher or lower than one. Analogous considerations can be made for Fig. 5, where v = 3. Differently from what was observed for a Gaussian beam [12

12. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]

], when a DNG particle was shifted towards the optical axis, here the total gradient forces now seems to push the DNG particle away from high intensity regions of the beam. This conclusion, however, cannot be extended to any relative refractive index. A full tridimensional plot of Fx as a function of both ρ0 and n for the zero-order Bessel beam used before, as shown in Fig. 6
Fig. 6 (a) Fx as a function of the displacement ρ 0 and n for a particle of radius a = 10λ, where λ = 1064 nm is the wavelength of our zero-order Bessel beam with an axicon angle θ = 0.0141 rad (spot of Δρ = 28.89 μm). (b) A contour plot of (a).
, reveals that it is also possible to have attractive forces towards high intensity regions (bright annular disks) of this multi-ringed beam even for DNG particles.

As Snell’s law is inverted, momentum transfer from the rays to a DNG particle leads to gradients forces whose repulsive/attractive profile is not seen for DPS particles. Depending upon the spatial distribution of power, P(r,θ,ϕ) and the radius a of the DNG particle, it can be pushed against or toward high intensity regions of the incident beam, regardless of |n| being higher or less than one.

In this paper we proposed a new application for double negative metamaterials in biomedical optics researches, using Bessel beams of arbitrary order. A full understanding of this new optical force profile can be obtained by extending the results of this work to include other regimes such as the Mie regime, making use of the generalized Lorenz-Mie theory for evaluating and studying the scattered fields. Obviously, scattering optical forces along the optical axis of the beam must also be revised, and other characteristics such as torque could also be investigated. This is currently in progress.

Although micro-spheres made of negative refractive index are not physically available yet, their technical feasibility may be just a matter of time. We believe that the manipulation of this kind of metamaterial particles using optical tweezers can be useful in biological applications, including new cancer treatments. Further studies may confirm this possibility.

Acknowledgements

The authors wish to thank Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under contracts 2009/54494-9 (L. A. Ambrosio’s post doctorate grant) and 2005/51689-2 (CePOF, Optics and Photonics Research Center), for supporting this work.

References and links

1.

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]

2.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991). [CrossRef] [PubMed]

3.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86(12), 4539–4543 (1989). [CrossRef] [PubMed]

4.

V. Emiliani, D. Cojoc, E. Ferrari, V. Garbin, C. Durieux, M. Coppey-Moisan, and E. Di Fabrizio, “Wave front engineering for microscopy of living cells,” Opt. Express 13(5), 1395–1405 (2005). [CrossRef] [PubMed]

5.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992). [CrossRef] [PubMed]

6.

G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7(6), 998–1007 (1990). [CrossRef]

7.

A. van der Horst, P. D. J. van Oostrum, A. Moroz, A. van Blaaderen, and M. Dogterom, “High trapping forces for high-refractive index particles trapped in dynamic arrays of counterpropagating optical tweezers,” Appl. Opt. 47(17), 3196–3202 (2008). [CrossRef] [PubMed]

8.

L. A. Ambrosio and H. E. Hernández-Figueroa, “Inversion of gradient forces for high refractive index particles in optical trapping,” Opt. Express 18(6), 5802–5808 (2010). [CrossRef] [PubMed]

9.

L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009). [CrossRef] [PubMed]

10.

L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express (to be published). [PubMed]

11.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001). [CrossRef]

12.

V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]

13.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]

14.

V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]

15.

L. A. Ambrosio, and H. E. Hernández-Figueroa are preparing a manuscript to be called “Integral localized approximation description of ordinary Bessel beams in the generalized Lorenz-Mie theory and application to optical forces.”

16.

L. A. Ambrosio, and H. E. Hernández-Figueroa, “Integral localized approximation description of v-th order Bessel beams in the generalized Lorenz-Mie theory and applications to optical trapping,” in Proceedings of PIERS2011in Marrakesh (to be published).

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(350.3618) Other areas of optics : Left-handed materials
(160.3918) Materials : Metamaterials
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: August 6, 2010
Revised Manuscript: October 22, 2010
Manuscript Accepted: October 28, 2010
Published: November 5, 2010

Virtual Issues
Vol. 6, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Leonardo A. Ambrosio and Hugo E. Hernández-Figueroa, "Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime," Opt. Express 18, 24287-24292 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24287


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References

  1. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef] [PubMed]
  2. R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991). [CrossRef] [PubMed]
  3. M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, and R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86(12), 4539–4543 (1989). [CrossRef] [PubMed]
  4. V. Emiliani, D. Cojoc, E. Ferrari, V. Garbin, C. Durieux, M. Coppey-Moisan, and E. Di Fabrizio, “Wave front engineering for microscopy of living cells,” Opt. Express 13(5), 1395–1405 (2005). [CrossRef] [PubMed]
  5. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992). [CrossRef] [PubMed]
  6. G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7(6), 998–1007 (1990). [CrossRef]
  7. A. van der Horst, P. D. J. van Oostrum, A. Moroz, A. van Blaaderen, and M. Dogterom, “High trapping forces for high-refractive index particles trapped in dynamic arrays of counterpropagating optical tweezers,” Appl. Opt. 47(17), 3196–3202 (2008). [CrossRef] [PubMed]
  8. L. A. Ambrosio and H. E. Hernández-Figueroa, “Inversion of gradient forces for high refractive index particles in optical trapping,” Opt. Express 18(6), 5802–5808 (2010). [CrossRef] [PubMed]
  9. L. A. Ambrosio and H. E. Hernández-Figueroa, “Trapping double negative particles in the ray optics regime using optical tweezers with focused beams,” Opt. Express 17(24), 21918–21924 (2009). [CrossRef] [PubMed]
  10. L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express (to be published). [PubMed]
  11. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001). [CrossRef]
  12. V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]
  13. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]
  14. V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
  15. L. A. Ambrosio, and H. E. Hernández-Figueroa are preparing a manuscript to be called “Integral localized approximation description of ordinary Bessel beams in the generalized Lorenz-Mie theory and application to optical forces.”
  16. L. A. Ambrosio, and H. E. Hernández-Figueroa, “Integral localized approximation description of v-th order Bessel beams in the generalized Lorenz-Mie theory and applications to optical trapping,” in Proceedings of PIERS2011in Marrakesh (to be published).

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