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Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 1, Iss. 5 — Dec. 1, 2010
  • pp: 1284–1301
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Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory

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


Biomedical Optics Express, Vol. 1, Issue 5, pp. 1284-1301 (2010)
http://dx.doi.org/10.1364/BOE.1.001284


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Abstract

Based on the generalized Lorenz-Mie theory (GLMT), this paper reveals, for the first time in the literature, the principal characteristics of the optical forces and radiation pressure cross-sections exerted on homogeneous, linear, isotropic and spherical hypothetical negative refractive index (NRI) particles under the influence of focused Gaussian beams in the Mie regime. Starting with ray optics considerations, the analysis is then extended through calculating the Mie coefficients and the beam-shape coefficients for incident focused Gaussian beams. Results reveal new and interesting trapping properties which are not observed for commonly positive refractive index particles and, in this way, new potential applications in biomedical optics can be devised.

© 2010 OSA

1. Introduction

Optical micromanipulation of molecules and biological organelles has revolutionized biomedical research and opened the way for an enormous quantity of promising studies evolving 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). [PubMed]

], chromosomes [2

2. 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,” in Proceedings of the National Academy of Science of the United States of America86, (1989), pp. 7914–7918.

], DNA [3

3. S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271(5250), 795–799 (1996). [PubMed]

], fungi [4

4. G. D. Wright, J. Arlt, W. C. K. Poon, and N. D. Read, “Experimentally manipulating fungi with optical tweezers,” Mycoscience 48, 15–19 (2007).

], human cells, general particles [5

5. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” in Proceedings of the National Academy of Science of the United States of America94, (1997), pp. 4853–4860.

,6

6. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [PubMed]

] and, maybe the most interesting, new proposals in cancer treatment [7

7. D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004). [PubMed]

].

Technological developments in the area of optical trapping are evident during the past forty years since the first experiments on Bell labs, by A. Ashkin. In a series of experiments, he put the theory behind tridimensional optical micromanipulation on solid grounds. Initially based on a pair of counter propagating moderately diverging Gaussian beams (2-beam traps) to capture randomly diffused small particles, he further improved the experimental setup to achieve levitation schemes (levitation traps), where the scattering forces exerted on a particle by one single vertical beam were used to cancel the effects of gravity. Finally, the adoption of one single focused beam paved the way for the systems now known as optical tweezers by allowing particles to be easily stretched and manipulated. A robust theory in damage-free optical traps of particles by using infrared laser beams is now available.

Optical traps are based on momentum transfer from the photons of the laser beam to the trapped particle, thus giving rise to axial and transverse forces that, in certain conditions, can pull them to high intensity regions of the beam. The relation between the repulsion/attraction profile and the relative refractive index nrel between the particle (refractive index np) and the surrounding medium (nm) in which it is immersed leads to a well known behavior: particles with nrel > 1 will always be directed towards high intensity regions of the incident beam, whereas for nrel < 1 the contrary can be observed: the particle will be directed away from these regions. For high refractive index particles (e.g., nrel > 3 or 4), however, this relation is no longer valid due to the prevailing of repulsive axial (scattering) forces [8

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). [PubMed]

].

Because of the increasing interest in metamaterial applications, especially among those with negative refractive index (NRI), it is plausible to ask ourselves what would happen to a NRI particle in an optical tweezers system. Would it be trapped in the same way as a positive refractive index (PRI) particle?

This special type of material is not a recent development. In fact, its properties are known since the final 60’s, when russian physicist Veselago predicted the hypothetical existence of NRI materials and consistently established a theoretical background for subsequent works [9

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

]. It took, however, more than thirty years for scientific community to realize the importance and the revolution behind Veselago’s medium, when the first experimental evidence of negative permittivity and permeability was published [10

10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [PubMed]

,11

11. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [PubMed]

].

This paper is the natural extension of [15

15. 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). [PubMed]

] and is organized as follows: section 2 reviews and significantly expands the analysis of [15

15. 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). [PubMed]

] for negative refractive index optical trapping using geometrical optics considerations, the main results concerning optical forces being then outlined; section 3 introduces the Mie coefficients and the equivalence between Snell’s law inversion and the differences in the Mie scattering coefficients when nrel goes from nrel > 0 to nrel < 0. A brief resume of the well-known generalized Lorenz-Mie theory (GLMT) is also presented. Finally, section 4 is devoted to discussing the application of the GLMT – and focused Gaussian beams – to NRI optical trapping. Transverse and longitudinal radiation pressure cross-sections (radiation pressure forces) are then numerically evaluated. In the final section, our conclusions are presented.

2. Ray optics

Suppose an arbitrary laser beam with wavelength λ, propagating along a medium of refractive index nm, impinges on an arbitrary NRI dielectric spherical particle with radius a. The particle is assumed to be homogeneous, isotropic and linear, with a refractive index np. In this section we consider that the ray optics condition, i.e., a >> λ, is tacitly satisfied, so that we may visualize the impinging beam as being composed of a set of infinite rays, all of them contributing to the total force exerted on the particle.

It is well-known that when a ray hits a positive refractive index particle with an incident angle θi, part of its incident power is reflected with the same incident angle and part of it is transmitted according to Snell’s law, i.e., sinθi = nrelsinθt, where θt is the transmission angle and nrel = np/nm. Subsequent reflections/refractions give rise to a multiple reflection/transmission diagram.

Due to the momentum transfer from the ray to the particle, one can analytically calculate the individual axial and transverse forces (Fz, z-directed, Fy, y-directed, according to the convention adopted in Fig. 1(c)
Fig. 1 Normalized (over nmP/c) individual transverse force Fy as a function of both θi and np for a circularly polarized ray over (a) a PRI and (c) a NRI particle. The difference observed between these two cases leads to new trapping phenomena for nrel < 0. (b) and (d) are the contour plots of (a) and (c), respectively.
of [15

15. 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). [PubMed]

]) by summing up the contribution of all rays leaving the particle, i.e., these individual forces are given by the contribution of the rays with power PR, PT 2, PT 2 R, PT 2 R 2, …, PT 2 Rm,…, P being the power of the original incident ray, R and T the Fresnel coefficients of reflection and transmission, respectively, their expressions depending upon the polarization of the incident beam, and m an integer number [16

16. 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). [PubMed]

].

For a NRI particle, however, due to the inversion of Snell’s law, the first transmitted ray into this particle is deflected with a “negative” angle θt, so that the individual transverse and axial forces must change accordingly (for instance, replace α by 2θi + 2θt and β by π + 2θt in Appendix I of [16

16. 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). [PubMed]

]. This leads to Eqs. (1) and (2) of [15

15. 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). [PubMed]

]).

Let us suppose that nrel = −1, so that R = 0 (T = 1) for a circularly polarized beam. Furthermore, assume that one of its infinite rays is a z-propagating ray impinging the NRI particle with an incident angle θi = 45°. For this particular ray, the first associated transmitted ray (with the same power as the incident ray, PT = P) is now y-directed, and the final ray exiting the particle propagates along –z, in a situation that resembles total reflection (R = 1, T = 0), except by the fact that the counter propagating final ray is spatially y-shifted from the incident ray. But conservation of linear momentum results in the same repulsive (away from the laser beam source) axial force. Of course, this situation would never happen for a PRI particle. Even though the axis of the incident ray does not coincide with the z-axis, being only parallel to it, there is no net transverse force. For rays with θi > 45°, Fy > 0 (i.e., repulsive), whereas Fy < 0 (attractive) for rays with θi < 45°. We can generalize this result by stating that, given a negative relative refractive index nrel, there is always an specific incident angle θi,s to which the following are true: Fy = 0 for θi = θi,s; Fy > 0 for θi > θi,s; and Fy < 0 for θi < θi,s. Axial forces are always repulsive for any single ray.

Figures 1(a) and 1(b) reveals the intensity of the axial force Fy for a PRI spherical particle as a function of both θi and np for a circularly polarized incident ray, where nm = 1.33 is assumed. According to the coordinate system adopted [15

15. 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). [PubMed]

], Fy > 0 (Fy < 0) corresponds to a repulsive (attractive) axial force. Two zero force lines, Fy = 0, have been highlighted in Fig. 1(b). The first one, Fy(nrel = 1), is the matched case where R = 0, T = 1, and θi = θt. The second zero-force line, Fy (nrel = nc), can be interpreted as the limit situation where, given an incident ray with an incident angle θi, an increase of the relative refractive index nrel (above a critical value nc) implies on the prevailing of the first reflected ray with power PR over all secondary rays of powers PT 2, PT 2 R and so on [10

10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [PubMed]

]. The scattering (axial) force Fz is always positive, i.e., repulsive, as expected [16

16. 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). [PubMed]

].

Figures 1(c) and 1(d) are plots of Fy as function of np and θi. Two zero force lines, representing the (nrel,θi = θi,s) points where Fy = 0, are now highlighted in Fig. 1(d). Contrary to Fig. 1(b), transverse forces do not become zero when |nrel| = 1. Note, furthermore, that in specifying nrel we are actually imposing the constraint between the force and the incidence angle and, as shown in Fig. 2
Fig. 2 Normalized (over nmP/c) individual transverse force Fy as a function of θi for (a) nrel = 1.2 (dotted) and −1.2 (solid) and (b) nrel = 0.8 (dotted) and −0.8 (solid). Force profiles can significantly vary for NRI and PRI particles possessing the same (in modulus) electromagnetic parameters. In (b), total reflection occurs for θi > 0.9273 rad.
for nrel = ± 1.2 and ± 0.8 (assuming the same values for the incident beam as before), the magnitude of the individual axial force exerted on a NRI particle can be significantly different from the PRI analogue.

A real situation can be used to illustrate how disparate these conclusions can be for both PRI and NRI optical trapping. Assume that the objective lens that focuses the Gaussian beam exiting the source has an associated numerical aperture θNA = 66°, and that 0.5 < nrel < 3. In this way, for all incident rays that compose the focused Gaussian beam, if the centre of the particle is located close to the focal point, θi < θNA = 66° and, according to Figs. 1(a) and 1(b), one can expect that an attractive (repulsive) transverse force will always occur whenever 1 < nrel < 3 (0 < nrel < 1).

But for nrel < 0 the situation is a little bit more involved, as now each ray can produce an attractive or a repulsive transverse force depending upon its incidence angle, so that, theoretically, the trapping properties for NRI particles depends upon the shape of the incident laser beam. For example, we could, in principle, design a particular laser beam such that, for an specific nrel, the particle will always be directed towards its high intensity regions, and another laser beam such that, for the same nrel, will always pull the particle towards nulls of intensity. This possibility is just impracticable for nrel > 0.

When considering a global coordinate system, each ray of our focused Gaussian beam impinges on the particle with a different angle relative to the optical axis [15

15. 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). [PubMed]

,16

16. 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). [PubMed]

]. Thus, we must consider total axial and transverse forces relative to a tridimensional global coordinate system. Although the details concerning the calculation of total axial and transverse forces are outside the scope of this paper, the reader is referred to [15

15. 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). [PubMed]

,16

16. 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). [PubMed]

] for further details.

The total transverse force exerted on a PRI particle is plotted in Fig. 3(a)
Fig. 3 (a) Ftransverse as a function of both nrel and r for a PRI particle under the influence of a focused Gaussian beam with w 0 = 1000 nm. The particle has a radius a = 10λ, where λ = 1064 nm is the wavelength of the beam. When nrel = 1, Ftransverse is always zero, as expected. (b) The contour plot of (a). Arbitrary units are adopted.
as function of both nrel and the distance r between the centre of the particle and the optical axis of a circularly polarized focused Gaussian beam (θNA = 66°) with a beam waist w 0 = 1 μm. The particle is assumed to be on a plane transverse to the optical axis and containing the focal point of the beam (the beam waist centre). The radius of the particle is a = 10λ, λ = 1064 nm. One can see attractive (negative) forces for all nrel just above 1 and repulsive (positive) forces for 0 < nrel < 1. Obviously, F transverse = 0 for nrel = 0, as expected. It is interesting to note the inversion of this transverse force from attractive to repulsive for high refractive index particles. In terms of ray optics, this is due to the fact that, as nrel increases, the power PR of the first reflected ray also increases, thus with a prevailing of repulsive forces [8

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). [PubMed]

]. As for the ripples, this is due to the vector nature of the optical force and, at least in the Mie regime, this phenomenon is associated with interference and resonance effects [17

17. A. B. Stilgoe, T. A. Nieminen, G. Knöener, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express 16(19), 15039–15051 (2008). [PubMed]

,18

18. Y. Hu, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Antireflection coating for improved optical trapping,” J. Appl. Phys. 103, 093119 (2008).

]. Figure 3(b) is a contour plot of Fig. 3(a).

Figure 4
Fig. 4 (a) Ftransverse as a function of both nrel and r for a NRI particle under the influence of the same laser beam and electromagnetic parameters as in Fig. 3. (b) The contour plot of (a). The same arbitrary units of Fig. 3 are adopted.
is the equivalent of Fig. 3 for a NRI particle with the same parameters as before. In this situation, the transverse total force profile can be interpreted as follows: regardless of nrel >˜ 1 or nrel <˜ 1, if the centre of the particle is close to the focus (beam waist centre), then it will always be attracted to the optical axis because transverse forces are attractive for low r. As |nrel| increases from zero, the range of possible r that still leads to attractive forces also increases but, above a certain distance, this force becomes repulsive, making optical trapping difficult to be achieved. There are no specific (constant) nrel that makes Ftransverse = 0, as in the PRI case (for nrel = 1). Finally, one can compare the amplitudes of Ftransverse in both figures. Attractive Ftransverse forces can be much stronger than these same forces acting on the equivalent PRI particle. For n = −1.31, Ftransverse|max = −7.24 (a.u.), whereas for n = 1.31, Ftransverse|max = −2.96 (a.u.), more than two times the expected force for a conventional PRI particle. Resonance effects are also observed for NRI particles with high refractive indices.

3. Review of the generalized Lorenz-Mie theory and its extension to negative refractive indices

The GLMT is an extension of the Lorenz-Mie theory [19

19. G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).

] for an incident beam of arbitrary shape and consists in expanding the incident electromagnetic field into a series of vector spherical harmonics, the coefficients of which being calculated by imposing the boundary conditions at the surface of the sphere and by making use of similar expressions for the scattered and internal fields [20

20. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).

22

22. G. Gouesbet, G. Gréhan, and B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).

].

In the framework of the Lorenz-Mie theory, in which the incident beam is a plane wave, the Mie scattering coefficients are known to be given as [23

23. C. F. Bohren, and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, John Wiley & Sons, 1983).

]
an=μnrelψn(nrelx)ψn(x)μpψn(x)ψn(nrelx)μnrelψn(nrelx)ξn(x)μpξn(x)ψn(nrelx)
(1)
bn=μpψn(nrelx)ψn(x)μnrelψn(x)ψn(nrelx)μpψn(nrelx)ξn(x)μnrelξn(x)ψn(nrelx)
(2)
where μp and μ are the permittivity of the particle and its surrounding medium, respectively, ψn and ξn are Riccati-Bessel functions, n (not to be confused with nrel, the relative refractive index) is an integer that ranges from 1 to + ∞ and x = ka is the size parameter of the particle, k being the wave number of the incident wave. The primes indicate derivatives with respect to the argument of the Riccati-Bessel functions. These relations should remain the same regardless of the scatter being a NRI or a PRI particle, because the tangential components of the fields at the interface (surface of the particle) are not affected by the negative refractive index of the particle [9

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

].

Let us turn our attention to Eqs. (1) and (2) and present, for the first time, the extension of the Mie scattering coefficients for a NRI particle. It is well known that resonance effects can be observed when their denominators approaches zero [23

23. C. F. Bohren, and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, John Wiley & Sons, 1983).

]. This can be mathematically represented by two transcendental equations: ψn(nrelx)/ψn(nrelx)=μpξn(x)/(μnrelξn(x)) and ψn(nrelx)/ψn(nrelx)=μnrelξn(x)/(μpξn(x)) for an and bn, respectively. Because nrel/μp is always positive, regardless of nrel being positive or negative, the right side of both equations remains unaffected whether we replace nrel by –nrel. On the other hand, the left side will be affected by this change in sign, and its slope will be reshaped, i.e., the size parameter associated to the peaks of an and bn will be different for a NRI and a PRI particle with the same (in modulus) refractive index. Note that this change of sign also affects both numerators in Eqs. (1) and (2), so that, in general, an and bn will differ significantly from the PRI case.

As an example, suppose a lossless and simple dielectric spherical PRI particle with nrel = 1.33. Figure 5
Fig. 5 Real (solid, red) and imaginary (dashed, blue) parts of the Mie scattering coefficient an as a function of the size parameter x for nrel = 1.33 and (a) n = 1, (b) n = 4, (c) n = 9 and (d) n = 16. In the framework of the GLMT, the coefficients an and bn modulates the phase and amplitude of the scattered fields.
shows Re(an) and Im(an) for n = 1, 4, 9 and 16 and 15 < x < 35, while Fig. 6
Fig. 6 Real (solid, red) and imaginary (dashed, blue) parts of the Mie coefficient an as a function of x for a NRI particle with nrel = −1.33 and (a) n = 1, (b) n = 4, (c) n = 9 and (d) n = 16. Different phase and amplitudes are observed in comparison with Fig. 5, so that the scattered fields will also be different.
is the equivalent of Fig. 5 for nrel = −1.33. We can conclude that, for a specific size parameter, i.e., given a PRI particle of fixed radius, the Mie scattering coefficients an will radically differ from the NRI analogue, the same being valid for the coefficients bn.

Thus, the scattered fields will have completely distinct spatial intensity distributions, as it was already pointed out by ray optics in section 2, where the fundamental difference between nrel > 0 and nrel < 0 lied solely in the inversion of Snell’s law. Here, the difference in the values of the Mie scattering coefficients accounts for an entire reshape (spatial intensity distribution) of the scattered fields due to its phase and amplitude contributions to each propagating mode.

As already pointed out, we can safely use Eqs. (1) and (2) for the scattering problem of a NRI spherical particle just in the same manner that it has being used so far for the conventional PRI case. Care should be exercised, however, by adequately replacing the permeability of the particle by its negative value whenever nrel < 0 or, equivalently, np < 0. This ensures that the results will effectively represent the correct scattered electromagnetic fields for a NRI particle. The other two possibilities, viz., (i) when nm < 0 and np > 0 or (ii) nm < 0 and np < 0 could also be analyzed, but we must consider that achieving a liquid medium for which nrel < 0 is even more challenging than achieving negative refractive index for a solid particle.

4. Radiation pressure calculations for NRI particles

Radiation pressure formulas for arbitrary laser beams incident on spherical particles are readily available from literature and are usually expressed as [29

29. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).

,30

30. K. F. Ren, G. Gréhan, and G. Gouesbet, “Symmetry relations in generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 11, 1812–1817 (1994).

]:
Cpr,z=λ2πn=1p=nn(1(n+1)2(n+1+|p|)!(n|p|)!Re[(an+an+12anan+1)gn,TMpgn+1,TMp+(bn+bn+12bnbn+1)gn,TEpgn+1,TEp]+p2n+1n2(n+1)2(n+|p|)!(n|p|)!Re[i(2anbnanbn)gn,TMpgn,TEp]),
(7)
C=λ22πp=1n=pm=p10((n+|p|)!(n|p|)![(Sm,np1+Sn,mp2Um,np12Un,mp)(1m2δm,n+11n2δn,m+1)+2n+1n2(n+1)2δn,m(Tm,np1Tn,mp2Vm,np1+2Vn,mp)]),
(8)
where
(Cpr,xCpr,y)=(Re(C)Im(C))
(9)
and the coefficients Un,mp,Vn,mp,Sn,mp,Tn,mp read as

Un,mp=anamgn,TMpgm,TMp+1+bnbmgn,TEpgm,TEp+1,
(10)
Vn,mp=ibnamgn,TEpgm,TMp+1ianbmgn,TMpgm,TEp+1,
(11)
Sn,mp=(an+am)gn,TMpgm,TMp+1+(bn+bm)gn,TEpgm,TEp+1,
(12)
Tn,mp=i(bn+am)gn,TEpgm,TMp+1i(an+bm)gn,TMpgm,TEp+1.
(13)

A Fortran code was developed for calculating Eqs. (7)(13) by using the Mie coefficients from Eqs. (1) and (2) and the associated BSC’s from Eqs. (5) and (6). This code is available under request. We used Eqs. (1) and (2) to generate the right-hand circularly polarized BSC’s, in accordance with [28

28. H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169–179 (1998).

], but we have not used any symmetry relation. This is more time consuming, but leads to a more ease-to-read program for those not familiar with the GLMT.

4.1. Longitudinal radiation pressure cross-section Cpr,z

During the first experiments on optical trapping, Ashkin noticed that negative values of Cpr,z were possible, depending on the longitudinal distance between the beam waist centre and the centre of the sphere. This was due the gradient of the intensity of the beam [16

16. 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). [PubMed]

]. In fact, full tridimensional traps demand Cpr,z = 0 at some specific point where Cpr,x and Cpr,y are also zero, thus providing stable equilibrium. For Cpr,z always non-negative, a point where Cpr,z would eventually be zero corresponds to a point of unstable equilibrium. It is still possible, however, to trap a particle even for Cpr,z ≥ 0, but this would require, for example, alternative schemes, such as levitation traps [31

31. A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).

35

35. A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19(5), 660–668 (1980). [PubMed]

]. Here, we shall not be concerned in obtaining an efficient tridimensional trap for NRI particles, but to examine some differences in their properties relative to PRI particles.

Let us consider that the beam waist centre of a right-hand circularly polarized Gaussian beam is located somewhere along the line (0,0,z 0). The beam has λ = 0.5 μm, beam waist w 0 = 5 μm and illuminates a particle of nrel = 1.5. Figure 7(a)
Fig. 7 Longitudinal radiation pressure cross-section Cpr,z as a function of z 0 for x 0 = y 0 = 0 for (a) nrel = 1.5 and (c) nrel = 1/1.33. (b) and (d) are the NRI analogues of (a) and (c), respectively.
shows Cpr,z as a function of z 0 for several diameters of the particle, viz., d = 5, 10, 20 and 40 μm. The profiles observed are just those expected and already studied by other authors [29

29. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).

].

As a second example, Fig. 8(a)
Fig. 8 (a) Cpr,z for several values of nrel assuming a PRI particle with radius a = 3.75 μm immersed on a focused Gaussian beam with λ = 0.3682 μm and w 0 = 1.8 μm. The same relative refractive indices were used in (b) for a NRI particle with the same radius as (a). The beam is shifted along its optical axis, i.e., x 0 = y 0 = 0.
is a reproduction of Fig. 8.11 from [36

36. K. R. Fen, Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques (Ph.D thesis, Faculté des Sciences de L’Université de Rouen, 1995).

] using the same parameters of the original work, i.e., a right-hand circularly polarized focused Gaussian beam with λ = 0.3682 μm, w 0 = 1.8 μm and a = 3.75 μm. Six values of nrel are used. The analogous NRI case with the same relative refractive indices (in modulus) is shown in Fig. 9(b)
Fig. 9 (a) Cpr,x for several diameters of a PRI particle with nrel = 1.5. The beam is shifted along x with y 0 = z 0 = 0, x 0 being the transverse distance between the optical axis and the centre of the particle. (b) The NRI analogue with nrel = −1.5.
for comparison. The following comments can be made regarding both figures: first, one notices negative longitudinal radiation pressure cross-section Cpr,z for positive nrel [Fig. 8(a)], whereas Cpr,z > 0 for all nrel < 0 [Fig. 8(b)]; second, the difference in magnitude (order of 102) of Cpr,z for the NRI and the PRI cases and the similitude of the slopes (superposed curves) for small variations of nrel when nrel < 0. Physical interpretations regarding the shape of the slopes of Cpr,z in Figs. 7 and 8 for nrel < 0, however, remains the same for nrel > 0 and can be found elsewhere [29

29. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).

].

It must be emphasized that negative longitudinal radiation pressure cross-section can also be achieved with lossless NRI spherical particles, including the case nrel = −1, which has been under intense investigation due to its incredible, new and until recently unimaginable possibility of applications, such as the famous perfect lens idealized by Pendry [37

37. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [PubMed]

]. So, consider as a final example a focused Gaussian beam with λ = 1064 nm (a typical injurious laser beam used in biological experiments) and spot w 0 = 1000 nm. These values coincide, at least for an on-axis beam, with the approximate theoretical limit of applicability of the localized beam model to GLMT calculations, as the parameter s = 1/kw 0 = 0.169 [26

26. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).

,27

27. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).

]. Calculations of Cpr,z for nrel = −1 for size parameters sp = 50, 100, 150 and 200 (viz., diameters of d ≈8.47, 16.93, 25.40 and 33.87 μm, respectively) are shown in Fig. 10
Fig. 10 (a) Cpr,z for a NRI particle with nrel = −1 and four different size parameters. The incident beam is a focused Gaussian beam with λ = 1064 nm and w 0 = 1.0 μm. The beam is shifted along its optical axis, i.e., x 0 = y 0 = 0.
, and we can clearly see negative values of Cpr,z and three points of stable equilibrium (indicated by arrows) concerning only the longitudinal pushing of the particle.

4.2. Transverse radiation pressure cross-sections Cpr,x and Cpr,y

If we focus our attention into Figs. 9(a), 11(a), 12(a), and 13(a), we will see the expected profile for a PRI particle under the influence of a right-hand circularly polarized focused Gaussian beam: whenever nrel > 1, the gradient of intensity of the beam generates attractive forces and, for 0 < nrel < 1, these forces are repulsive. But there is something more to which we should look at carefully, regarding the optical regime and the vector nature of the incident beam. First, the amplitudes observed in Figs. 9(a) and 12(a) and also in Figs. 11(a) and 13(a) are close to each other, so that we should ask ourselves if this is correct. Looking back at the literature, Gouesbet et al [29

29. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).

] found different amplitudes for these same parameters. But a moment thought would lead us to conclude that this should not happen because of the symmetry of the incident beam, its polarization (circular) and the electromagnetic properties and the ratio d/λ of all particles (e.g., for d = 5 μm, d/λ = 10, which would ultimately define the lower limit of applicability of geometrical optics). Obviously, if a linearly polarized beam is used, then Cpr,x and Cpr,y differ in magnitude, depending upon the relative direction of displacement between the beam waist centre and the centre of the particle. These considerations are immediately extended to the NRI case, viz., Figs. 9(b), 11(b), 12(b), and 13(b).

Because linear momentum is transferred to a NRI particle in a slightly different way, the Mie scattering coefficients alter the values of the longitudinal and transverse radiation pressure cross-sections in Eqs. (7) and (9), respectively, thus leading to new force profiles. Thus, the GLMT reflects the new scattered field for a NRI particle.

When the relative refractive index is −1.5, one can see from Figs. 9(b) and 12(b) that both repulsive and attractive transverse forces can act on the particle, depending on the relative distance x 0 between the beam waist centre and the centre of the particle. Because d/λ is always higher than or equal to 10, a qualitative explanation can be done, based on geometrical optics considerations. First, consider d = 40 μm. This implies d/w 0 = 8, so that, for short relative distances x 0, the rays impinge the particle with small incidence angles, thus giving rise to attractive forces. This can be better appreciated by considering individual transverse forces analogous to Fig. 2 but for nrel = −1.5, graphically represented by a dotted curve in Fig. 14
Fig. 14 Normalized (over nmP/c) individual transverse force Ftransverse as a function of θi for both nrel = −1.5 (dotted) and −0.5 (solid). For the last case, total reflection occurs for θi > 0.5236 rad.
. As the beam waist centre gets close to x 0 = 17 μm, i.e., close to the radius of the particle, repulsive individual transverse forces prevails, as expected from Fig. 14 for high incidence angles. Notice, however, that the peak of this repulsive force (Cpr,x ≈2.70 × 10−12 m2 at x 0 ≈19.41 μm) is relatively low compared to the peak observed in the region of attractive force (Cpr,x ≈14.69 × 10−11 m2 at x 0 ≈9.11 μm). As d/w 0 decreases, so does the amplitude of the transverse force. When d/w 0 = 1 (e.g., d = 5 μm), repulsive forces will take place for x 0 > 7.52 μm but, due to their low amplitude, this is not readily seen in Fig. 9(b). For Figs. 11(b) and 13(b), we must further consider the fact that, above θi > 0.5236 rad, an incident ray will suffer total reflection and, according to the solid curve in Fig. 14, higher repulsive forces can be expected. This is, in fact, true, as shown for Cpr,x and Cpr,y in Figs. 11(b) and 13(b), respectively. Finally, because of total reflection, it is not always possible to trap NRI particles with −1 < nrel < 0. For the parameters used, NRI particles with nrel = −0.5 and d = 5 or 10 μm would always be directed away from the beam waist centre. Thus, as |nrel| decreases, repulsive forces prevail. This is illustrated in Fig. 15
Fig. 15 Cpr,x as a function of the displacement x 0 and the relative refractive index nrel for a NRI particle with d = 40 μm.
for six different values of nrel.

5. Conclusions

Forces and radiation pressure cross-sections were systematically analyzed for lossless negative refractive index spherical and simple particles. It was shown, for the first time in the literature, both by ray optics and by adopting the generalized Lorenz-Mie theory with the integral localized approximation, that the forces and radiation pressure cross-sections behave quite differently due to the new linear momentum transfer characteristic that takes place for NRI particles.

In the ray optics, the inversion of Snell’s law accounts for individual forces whose dependence with the incidence angle reveals new capabilities in trapping NRI particles, because both attractive and repulsive forces can happen depending upon how the ray impinges the particle. This characteristic is just unrealizable for PRI particles, as the transmission angle is always “positive”. In the framework of the generalized Lorenz-Mie theory, a NRI particle will present different force profiles when compared to the conventional PRI optical trapping because of the new scattered field, theoretically represented by the Mie scattering coefficients.

Negative longitudinal radiation pressure cross-sections can also be obtained in the NRI case by suitably choosing the relative refractive index of the particle and its dimensions relative to the parameters of the incident laser beam. It is not possible, however, to formulate an explanation of NRI optical trapping based solely on the relative refractive index of the particle and the gradient intensity of the beam in the optical regimes of this paper.

The fact that repulsive transverse forces prevail over attractive ones as the relative distance between the beam waist centre and the centre of the NRI particle may turn NRI optical trapping into an experimentally more difficult task, because care would have to be exercised in spatially manipulating the laser beam so that an effective optical trap is produced.

All results presented here were obtained for a (right hand) circularly polarized focused Gaussian beam. Other types of polarization could be implemented, as well as other laser beams such as Bessel and Laguerre-Gaussian beams. In fact, the whole theory of optical trapping, including resonance effects and torque properties, must be revised for NRI optical trapping. These studies are currently under investigation.

The question of how to experimentally achieve a homogeneous, linear, isotropic, lossless spherical negative refractive index material is still open. Resonant tridimensional metamaterial structures possessing negative refractive index have been theoretically demonstrated, but the difficulty of realizing theoretical and experimental non-resonant tridimensional NRI materials may be overcome in the forthcoming years.

Acknowledgments

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). [PubMed]

2.

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,” in Proceedings of the National Academy of Science of the United States of America86, (1989), pp. 7914–7918.

3.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271(5250), 795–799 (1996). [PubMed]

4.

G. D. Wright, J. Arlt, W. C. K. Poon, and N. D. Read, “Experimentally manipulating fungi with optical tweezers,” Mycoscience 48, 15–19 (2007).

5.

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” in Proceedings of the National Academy of Science of the United States of America94, (1997), pp. 4853–4860.

6.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [PubMed]

7.

D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004). [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). [PubMed]

9.

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

10.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [PubMed]

11.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [PubMed]

12.

N. Engheta and R. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).

13.

N. Engheta and R. Ziolkowski, Metamaterials – Physics and Engineering Explorations (IEEE press, Wiley-Interscience, John Wiley & Sons, 2006).

14.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (IEEE press, Wiley-Interscience, John Wiley & Sons, 2006).

15.

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). [PubMed]

16.

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). [PubMed]

17.

A. B. Stilgoe, T. A. Nieminen, G. Knöener, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express 16(19), 15039–15051 (2008). [PubMed]

18.

Y. Hu, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Antireflection coating for improved optical trapping,” J. Appl. Phys. 103, 093119 (2008).

19.

G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).

20.

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).

21.

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).

22.

G. Gouesbet, G. Gréhan, and B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).

23.

C. F. Bohren, and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, John Wiley & Sons, 1983).

24.

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, 998–1007 (1990).

25.

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. 37(19), 4218–4225 (1998). [PubMed]

26.

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).

27.

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).

28.

H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169–179 (1998).

29.

K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).

30.

K. F. Ren, G. Gréhan, and G. Gouesbet, “Symmetry relations in generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 11, 1812–1817 (1994).

31.

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).

32.

A. Ashkin and J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24, 586–588 (1974).

33.

A. Ashkin and J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. 28, 333–335 (1976).

34.

A. Ashkin and J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).

35.

A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19(5), 660–668 (1980). [PubMed]

36.

K. R. Fen, Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques (Ph.D thesis, Faculté des Sciences de L’Université de Rouen, 1995).

37.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [PubMed]

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(290.4020) Scattering : Mie theory
(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 Traps, Manipulation, and Tracking

History
Original Manuscript: August 6, 2010
Revised Manuscript: October 13, 2010
Manuscript Accepted: October 17, 2010
Published: November 4, 2010

Citation
Leonardo A. Ambrosio and Hugo 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 1, 1284-1301 (2010)
http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-5-1284


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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). [PubMed]
  2. 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,” in Proceedings of the National Academy of Science of the United States of America86, (1989), pp. 7914–7918.
  3. S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271(5250), 795–799 (1996). [PubMed]
  4. G. D. Wright, J. Arlt, W. C. K. Poon, and N. D. Read, “Experimentally manipulating fungi with optical tweezers,” Mycoscience 48, 15–19 (2007).
  5. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” in Proceedings of the National Academy of Science of the United States of America94, (1997), pp. 4853–4860.
  6. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004). [PubMed]
  7. D. P. O’Neal, L. R. Hirsch, N. J. Halas, J. D. Payne, and J. L. West, “Photo-thermal tumor ablation in mice using near infrared-absorbing nanoparticles,” Cancer Lett. 209(2), 171–176 (2004). [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). [PubMed]
  9. V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Usp. 4, 509–514 (1968).
  10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [PubMed]
  11. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [PubMed]
  12. N. Engheta and R. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1535–1556 (2005).
  13. N. Engheta and R. Ziolkowski, Metamaterials – Physics and Engineering Explorations (IEEE press, Wiley-Interscience, John Wiley & Sons, 2006).
  14. C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications (IEEE press, Wiley-Interscience, John Wiley & Sons, 2006).
  15. 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). [PubMed]
  16. 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). [PubMed]
  17. A. B. Stilgoe, T. A. Nieminen, G. Knöener, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express 16(19), 15039–15051 (2008). [PubMed]
  18. Y. Hu, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Antireflection coating for improved optical trapping,” J. Appl. Phys. 103, 093119 (2008).
  19. G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
  20. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
  21. B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
  22. G. Gouesbet, G. Gréhan, and B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
  23. C. F. Bohren, and D. R. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, John Wiley & Sons, 1983).
  24. 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, 998–1007 (1990).
  25. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. 37(19), 4218–4225 (1998). [PubMed]
  26. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
  27. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
  28. H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. 155, 169–179 (1998).
  29. K. F. Ren, G. Gréhan, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
  30. K. F. Ren, G. Gréhan, and G. Gouesbet, “Symmetry relations in generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 11, 1812–1817 (1994).
  31. A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
  32. A. Ashkin and J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24, 586–588 (1974).
  33. A. Ashkin and J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. 28, 333–335 (1976).
  34. A. Ashkin and J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).
  35. A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19(5), 660–668 (1980). [PubMed]
  36. K. R. Fen, Diffusion des Faisceaux Feuille Laser par une Particule Sphérique et Applications aux Ecoulements Diphasiques (Ph.D thesis, Faculté des Sciences de L’Université de Rouen, 1995).
  37. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [PubMed]

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