## Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory |

Biomedical Optics Express, Vol. 1, Issue 5, pp. 1284-1301 (2010)

http://dx.doi.org/10.1364/BOE.1.001284

Acrobat PDF (2219 KB)

### 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

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

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]

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]

*n*between the particle (refractive index

_{rel}*n*) and the surrounding medium (

_{p}*n*) in which it is immersed leads to a well known behavior: particles with

_{m}*n*> 1 will always be directed towards high intensity regions of the incident beam, whereas for

_{rel}*n*< 1 the contrary can be observed: the particle will be directed away from these regions. For high refractive index particles (e.g.,

_{rel}*n*> 3 or 4), however, this relation is no longer valid due to the prevailing of repulsive axial (scattering) forces [8

_{rel}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]

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]

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]

*n*= 1 (in the NRI analogue,

_{rel}*n*= −1) does not play such a significant role as it does for PRI particles.

_{rel}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]

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]

*n*goes from

_{rel}*n*> 0 to

_{rel}*n*< 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.

_{rel}## 2. Ray optics

*λ*, propagating along a medium of refractive index

*n*, impinges on an arbitrary NRI dielectric spherical particle with radius

_{m}*a*. The particle is assumed to be homogeneous, isotropic and linear, with a refractive index

*n*. In this section we consider that the ray optics condition, i.e.,

_{p}*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.

*θ*, 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., sin

_{i}*θ*=

_{i}*n*sin

_{rel}*θ*, where

_{t}*θ*is the transmission angle and

_{t}*n*=

_{rel}*n*/

_{p}*n*. Subsequent reflections/refractions give rise to a multiple reflection/transmission diagram.

_{m}*F*,

_{z}*z*-directed,

*F*,

_{y}*y*-directed, according to the convention adopted in Fig. 1(c) of [15

**17**(24), 21918–21924 (2009). [PubMed]

*PR*,

*PT*

^{2},

*PT*

^{2}

*R*,

*PT*

^{2}

*R*

^{2}, …,

*PT*

^{2}

*R*,…,

^{m}*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]

*θ*, so that the individual transverse and axial forces must change accordingly (for instance, replace

_{t}*α*by 2

*θ*+ 2

_{i}*θ*and

_{t}*β*by π + 2

*θ*in Appendix I of [16

_{t}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**(24), 21918–21924 (2009). [PubMed]

*n*= −1, so that

_{rel}*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

*θ*= 45°. For this particular ray, the first associated transmitted ray (with the same power as the incident ray,

_{i}*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

*θ*> 45°,

_{i}*F*> 0 (i.e., repulsive), whereas

_{y}*F*< 0 (attractive) for rays with

_{y}*θ*< 45°. We can generalize this result by stating that, given a negative relative refractive index

_{i}*n*, there is always an specific incident angle

_{rel}*θ*to which the following are true:

_{i,s}*F*= 0 for

_{y}*θ*=

_{i}*θ*;

_{i,s}*F*> 0 for

_{y}*θ*>

_{i}*θ*; and

_{i,s}*F*< 0 for

_{y}*θ*<

_{i}*θ*. Axial forces are always repulsive for any single ray.

_{i,s}*F*for a PRI spherical particle as a function of both

_{y}*θ*and

_{i}*n*for a circularly polarized incident ray, where

_{p}*n*= 1.33 is assumed. According to the coordinate system adopted [15

_{m}**17**(24), 21918–21924 (2009). [PubMed]

*F*> 0 (

_{y}*F*< 0) corresponds to a repulsive (attractive) axial force. Two zero force lines,

_{y}*F*= 0, have been highlighted in Fig. 1(b). The first one,

_{y}*F*(

_{y}*n*= 1), is the matched case where

_{rel}*R*= 0,

*T*= 1, and

*θ*=

_{i}*θ*. The second zero-force line,

_{t}*F*(

_{y}*n*=

_{rel}*n*), can be interpreted as the limit situation where, given an incident ray with an incident angle

_{c}*θ*, an increase of the relative refractive index

_{i}*n*(above a critical value

_{rel}*n*) implies on the prevailing of the first reflected ray with power

_{c}*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]

*F*is always positive, i.e., repulsive, as expected [16

_{z}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]

*F*as function of

_{y}*n*and

_{p}*θ*. Two zero force lines, representing the (

_{i}*n*,

_{rel}*θ*=

_{i}*θ*) points where

_{i,s}*F*= 0, are now highlighted in Fig. 1(d). Contrary to Fig. 1(b), transverse forces do not become zero when |

_{y}*n*| = 1. Note, furthermore, that in specifying

_{rel}*n*we are actually imposing the constraint between the force and the incidence angle and, as shown in Fig. 2 for

_{rel}*n*= ± 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.

_{rel}*θ*= 66°, and that 0.5 <

_{NA}*n*< 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,

_{rel}*θ*<

_{i}*θ*= 66° and, according to Figs. 1(a) and 1(b), one can expect that an attractive (repulsive) transverse force will always occur whenever 1 <

_{NA}*n*< 3 (0 <

_{rel}*n*< 1).

_{rel}*n*< 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

_{rel}*n*, the particle will always be directed towards its high intensity regions, and another laser beam such that, for the same

_{rel}*n*, will always pull the particle towards nulls of intensity. This possibility is just impracticable for

_{rel}*n*> 0.

_{rel}**17**(24), 21918–21924 (2009). [PubMed]

**61**(2), 569–582 (1992). [PubMed]

**17**(24), 21918–21924 (2009). [PubMed]

**61**(2), 569–582 (1992). [PubMed]

*n*and the distance

_{rel}*r*between the centre of the particle and the optical axis of a circularly polarized focused Gaussian beam (

*θ*= 66°) with a beam waist

_{NA}*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*= 10

*λ*,

*λ*= 1064 nm. One can see attractive (negative) forces for all

*n*just above 1 and repulsive (positive) forces for 0 <

_{rel}*n*< 1. Obviously,

_{rel}**F**

*= 0 for*

_{transverse}*n*= 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

_{rel}*n*increases, the power

_{rel}*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]

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]

*n*

_{rel}*n*

_{rel}*r*. As |

*n*| increases from zero, the range of possible

_{rel}*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)

*n*that makes

_{rel}*F*= 0, as in the PRI case (for

_{transverse}*n*= 1). Finally, one can compare the amplitudes of

_{rel}*F*in both figures. Attractive

_{transverse}*F*forces can be much stronger than these same forces acting on the equivalent PRI particle. For

_{transverse}*n*= −1.31,

*F*|

_{transverse}_{max}= −7.24 (a.u.), whereas for

*n*= 1.31,

*F*|

_{transverse}_{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

*μ*and

_{p}*μ*are the permittivity of the particle and its surrounding medium, respectively,

*n*(not to be confused with

*n*, the relative refractive index) is an integer that ranges from 1 to + ∞ and

_{rel}*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].

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]

*E*and

_{r}*H*, a localization operator

_{r}*n*<

*m*<

*n*. Note that, in Eqs. (3) and (4), spherical coordinates (

*r*,

*θ*,

*ϕ*) are assumed and, therefore, the incident beam must be changed to a spherical coordinate system accordingly (in this paper, we tacitly assume that the reader is familiarized with both the mathematical background and the notation adopted herein. As the BSC’s are not altered whether the particle is of NRI or PRI nature, we shall not go into further details (for additional information see, e.g., [20–22,24,25

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]

*z*-propagating linearly

*x*-polarized beam, as [26,27]: where

*x*

_{0},

*y*

_{0},

*z*

_{0}) of the beam (relative distance between the beam waist centre and the centre of the spherical particle) and its parameters in an spherical coordinate system [27].

*δ*is the Kronecker delta symbol. For circularly polarized focused Gaussian beams, the BSC’s can be easily evaluated by considering symmetry relations [28].

_{ij}*a*and

_{n}*b*, respectively. Because

_{n}*n*/

_{rel}*μ*is always positive, regardless of

_{p}*n*being positive or negative, the right side of both equations remains unaffected whether we replace

_{rel}*n*by –

_{rel}*n*. 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

_{rel}*a*and

_{n}*b*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,

_{n}*a*and

_{n}*b*will differ significantly from the PRI case.

_{n}*n*= 1.33. Figure 5 shows

_{rel}*Re*(

*a*) and

_{n}*Im*(

*a*) for

_{n}*n*= 1, 4, 9 and 16 and 15 <

*x*< 35, while Fig. 6 is the equivalent of Fig. 5 for

*n*= −1.33. We can conclude that, for a specific size parameter, i.e., given a PRI particle of fixed radius, the Mie scattering coefficients

_{rel}*a*will radically differ from the NRI analogue, the same being valid for the coefficients

_{n}*b*.

_{n}*n*> 0 and

_{rel}*n*< 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.

_{rel}*n*< 0 or, equivalently,

_{rel}*n*< 0. This ensures that the results will effectively represent the correct scattered electromagnetic fields for a NRI particle. The other two possibilities,

_{p}*viz*., (i) when

*n*< 0 and

_{m}*n*> 0 or (ii)

_{p}*n*< 0 and

_{m}*n*< 0 could also be analyzed, but we must consider that achieving a liquid medium for which

_{p}*n*< 0 is even more challenging than achieving negative refractive index for a solid particle.

_{rel}*n*< 0 and

_{m}*n*> 0 or

_{p}*n*> 0 and

_{m}*n*< 0 (or even replacing conditions

_{p}*n*> 0 and

_{m}*n*> 0 by

_{p}*n*< 0 and

_{m}*n*< 0) leads to the same Mie scattering coefficients and, therefore, to the same force profiles. Accordingly, in the ray optics approach, these conditions imply in the same multiple reflection/refraction diagrams observed in previous works [15

_{p}**17**(24), 21918–21924 (2009). [PubMed]

**61**(2), 569–582 (1992). [PubMed]

## 4. Radiation pressure calculations for NRI particles

### 4.1. Longitudinal radiation pressure cross-section C_{pr,z}

*C*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

_{pr,z}**61**(2), 569–582 (1992). [PubMed]

*C*= 0 at some specific point where

_{pr,z}*C*and

_{pr,x}*C*are also zero, thus providing stable equilibrium. For

_{pr,y}*C*always non-negative, a point where

_{pr,z}*C*would eventually be zero corresponds to a point of unstable equilibrium. It is still possible, however, to trap a particle even for

_{pr,z}*C*≥ 0, but this would require, for example, alternative schemes, such as levitation traps [31–35

_{pr,z}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]

*n*= −1.5. Figure 7(b) presents the new

_{rel}*C*profiles. Although the locations of maxima of

_{pr,z}*C*occurs at about the same position

_{pr,z}*z*

_{0}, we can see the disparate values in magnitude for a specific

*d*when compared to the PRI case in Fig. 7(a). For example, for

*d*= 40 μm, maxima of

*C*

_{pr,z}_{(NRI)}can be seen at

*z*

_{0}≈5.64 × 10

^{−4}m and −5.45 × 10

^{−4}m, whereas maxima of

*C*

_{pr,z}_{(PRI)}occur at

*z*

_{0}≈6.04 × 10

^{−4}m and −5.84 × 10

^{−4}m. The ratio of these maxima are

*C*

_{pr,z}_{(NRI)}|

_{z}_{0 ≈5.64 × 10-4 m}/

*C*

_{pr,z}_{(PRI)}|

_{z}_{0 ≈6.04 × 10-4 m}≈4.31 and

*C*

_{pr,z}_{(NRI)}|

_{z}_{0 ≈-5.45 × 10-4 m}/

*C*

_{pr,z}_{(PRI)}|

_{z}_{0 ≈-5.84 × 10-4 m}≈4.17, representing a much stronger longitudinal radiation pressure for the NRI particle. It must be emphasized that the longitudinal radiation pressure cross-section can be significantly different for the NRI case when compared with the conventional PRI analogue. Again, this is due to the numerical differences in the Mie coefficients observed in the previous section or, equivalently, due to the distinct reflection/transmission diagram for an incident ray in ray optics.

*λ*= 0.3682 μm,

*w*

_{0}= 1.8 μm and

*a*= 3.75 μm. Six values of

*n*are used. The analogous NRI case with the same relative refractive indices (in modulus) is shown in Fig. 9(b) for comparison. The following comments can be made regarding both figures: first, one notices negative longitudinal radiation pressure cross-section

_{rel}*C*for positive

_{pr,z}*n*[Fig. 8(a)], whereas

_{rel}*C*> 0 for all

_{pr,z}*n*< 0 [Fig. 8(b)]; second, the difference in magnitude (order of 10

_{rel}^{2}) of

*C*for the NRI and the PRI cases and the similitude of the slopes (superposed curves) for small variations of

_{pr,z}*n*when

_{rel}*n*< 0. Physical interpretations regarding the shape of the slopes of

_{rel}*C*in Figs. 7 and 8 for

_{pr,z}*n*< 0, however, remains the same for

_{rel}*n*> 0 and can be found elsewhere [29].

_{rel}*n*= −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

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

*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 [26,27]. Calculations of

*C*for

_{pr,z}*n*= −1 for size parameters

_{rel}*s*= 50, 100, 150 and 200 (

_{p}*viz*., diameters of

*d*≈8.47, 16.93, 25.40 and 33.87 μm, respectively) are shown in Fig. 10 , and we can clearly see negative values of

*C*and three points of stable equilibrium (indicated by arrows) concerning only the longitudinal pushing of the particle.

_{pr,z}### 4.2. Transverse radiation pressure cross-sections C_{pr,x} and C_{pr,y}

*n*> 1, an optical trap is to be expected with the particle localized at points of stable equilibrium or, as an another perspective, at high intensity regions of the laser beam. On contrary, if

_{rel}*n*< 1, then it will invariably be directed away from these high intensity regions. In the case of focused Gaussian beams, this last situation can be interpreted as a limitation in achieving an efficient trap. There are, of course, other types of beams (

_{rel}*viz*., Bessel beams) where a set of spatial positions (low intensity regions of the beam) exists in which a particle possessing

*n*< 1 can still be trapped, again allowing three- or two-dimensional manipulation of particles.

_{rel}*n*. We can conclude, by recalling the previous analysis of section 2, that the relative distance between the focal point and the centre of the particle is a determinant variable in calculating axial forces. While these forces seem to be attractive for short relative distances, as the particle goes away from the focus, they may become repulsive, a behavior which would never be expected for a PRI particle but that can be easily predicted for a NRI particle using simple ray optics considerations. Thus, we cannot try to use only gradient intensity considerations (together with the well-known repulsive/attractive critical limits

_{p}*n*< 1 and

_{rel}*n*> 1) when studying NRI particles immersed in a positive refractive index medium.

_{rel}*n*> 1, the gradient of intensity of the beam generates attractive forces and, for 0 <

_{rel}*n*< 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

_{rel}*et al*[29] 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

*C*and

_{pr,x}*C*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,

_{pr,y}*viz*., Figs. 9(b), 11(b), 12(b), and 13(b).

*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

*n*= −1.5, graphically represented by a dotted curve in Fig. 14 . As the beam waist centre gets close to

_{rel}*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 (

*C*≈2.70 × 10

_{pr,x}^{−12}m

^{2}at

*x*

_{0}≈19.41 μm) is relatively low compared to the peak observed in the region of attractive force (

*C*≈14.69 × 10

_{pr,x}^{−11}m

^{2}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

*θ*> 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

_{i}*C*and

_{pr,x}*C*in Figs. 11(b) and 13(b), respectively. Finally, because of total reflection, it is not always possible to trap NRI particles with −1 <

_{pr,y}*n*< 0. For the parameters used, NRI particles with

_{rel}*n*= −0.5 and

_{rel}*d*= 5 or 10 μm would always be directed away from the beam waist centre. Thus, as |

*n*| decreases, repulsive forces prevail. This is illustrated in Fig. 15 for six different values of

_{rel}*n*.

_{rel}## 5. Conclusions

## Acknowledgments

## References and links

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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 |

4. | G. D. Wright, J. Arlt, W. C. K. Poon, and N. D. Read, “Experimentally manipulating fungi with optical tweezers,” Mycoscience |

5. | A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” in |

6. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

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. |

8. | L. A. Ambrosio and H. E. Hernández-Figueroa, “Inversion of gradient forces for high refractive index particles in optical trapping,” Opt. Express |

9. | V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,” Sov. Phys. Usp. |

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. |

11. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

12. | N. Engheta and R. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microw. Theory Tech. |

13. | N. Engheta and R. Ziolkowski, |

14. | C. Caloz and T. Itoh, |

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 |

16. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. |

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 |

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

19. | G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. |

20. | G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) |

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

22. | G. Gouesbet, G. Gréhan, and B. Maheu, “Expressions to compute the coefficients g 19, 35–48 (1988). |

23. | C. F. Bohren, and D. R. Huffmann, |

24. | G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients 7, 998–1007 (1990). |

25. | K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized lorenz-mie theory,” Appl. Opt. |

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 |

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 |

28. | H. Polaert, G. Gréhan, and G. Gouesbet, “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun. |

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. |

30. | K. F. Ren, G. Gréhan, and G. Gouesbet, “Symmetry relations in generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A |

31. | A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. |

32. | A. Ashkin and J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. |

33. | A. Ashkin and J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. |

34. | A. Ashkin and J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. |

35. | A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. |

36. | K. R. Fen, |

37. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

**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

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