## Trapping double negative particles in the ray optics regime using optical tweezers with focused beams

Optics Express, Vol. 17, Issue 24, pp. 21918-21924 (2009)

http://dx.doi.org/10.1364/OE.17.021918

Acrobat PDF (590 KB)

### Abstract

The capabilities of optical tweezers to trap DNG (double negative) spherical particles, with both negative permittivity and permeability, are explored in detail by analyzing some interesting theoretical features not seeing in conventional DPS (double positive) particles possessing positive refractive index. The ray optics regime is adopted and, although this regime is quite simple and limited, its validity is already known and tested for DPS particles such as biological cells and molecules trapped by highly focused beams. Simulation results confirm that even for ray optics, DNG particles present unusual and interesting trapping characteristics.

© 2009 Optical Society of America

## 1. Introduction

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**, 156–159 (
1970). [CrossRef]

2. A. Ashkin, “Atomic-beam deflection by resonance-radiation pressure,” Phys. Rev. Lett. **24**, 1321–1324 (
1970). [CrossRef]

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

4. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (
1986). [CrossRef] [PubMed]

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

6. 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**, 505–510 (
1991). [CrossRef] [PubMed]

7. 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**, 7914–7918 (
1989). [CrossRef]

8. V. Emiliani*et al*., “Wave front engineering for microscopy of living cells,” Opt. Express **13**, 1395–1405 (
2005). [CrossRef] [PubMed]

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

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 (
2000). [CrossRef]

11. J. B. Pendry, “Negative Refraction Makes Perfect Lens,” Phys. Rev. Lett. **85**, 18 (
2000). [CrossRef]

## 2. Theoretical analysis

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

*f*is the focus of the beam, where all incident rays would converge in the absence of the sphere.

**F**

_{1}and

**F**

_{2}for

*n*

_{1}>

*n*

_{2}and the particle is directed away from their axis; for

*n*

_{1}<

*n*

_{2}, these forces become attractive, and the particle tends to be aligned with the optical axis.

*F*to be the

_{g}*x*-component of the vector force that points in a direction perpendicular to the axis of the ray, and the scattering force

*F*to be the component along this axis. As a result of the negative refractive angle, we find

_{S}*c*is the speed of light in vacuum,

*R*and

*T*are the Fresnel coefficients of reflection and transmission, respectively. Due to the difference of 2

*θ*for the first transmitted ray, in Eqs. (1) and (2) there is a change in sign in the argument of both cosine and sine of (2

_{i}*θ*+2

_{i}*θ*) when compared to conventional particles, resulting in different forces [14

_{t}14. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**, 569–582 (
1992). [CrossRef] [PubMed]

*F*and

_{g}*F*are plotted as functions of

_{S}*θ*, supposing a highly focused beam with a numerical aperture of 66°, typical of the microscopes used in experimental setups. A circularly polarized beam has been assumed, and we have imposed

_{i}*n*

_{1}=1.33 for the medium, |

*n*

_{2}|=1.62 (Figs. 2(a) and 2(b)) or |

*n*

_{2}|=1.21 (Figs. 2(c) and 2(d)).

*F*(solid lines) is directed toward the ray axis – tending to bring it closer to the ray itself – for

_{g}*n*

_{2}>

*n*

_{1}, or away from this axis – repulsive force – for

*n*

_{2}<

*n*

_{1}. The scattering force (dashed lines) is always positive, growing in magnitude for higher angles of incidence, as expected.

*θ*≈25°. One important characteristic of DNG particles is that, at least in the ray optics regime, gradient forces do not change sign if we change from |

_{i}*n*

_{2}|<

*n*

_{1}to |

*n*

_{2}|>

*n*

_{1}.

*n*

_{2}|<

*n*

_{1}, one can conclude by looking at Figs. 2(c) and 2(d) that, as the beam has its maximum intensity along its optical axis, the forces exerted in the particle will be repulsive when its index

*n*

_{2}is positive and, for a metamaterial, although the scattering force is attractive, the gradient component of the total force for a ray will be either attractive or repulsive, depending on the angle of incidence.

### 2.1Total forces as functions of the angle γ

**F**when the converging cone of the beam is restricted within 0≤

*θ*≤

*θ*

_{max},

*θ*

_{max}=66° being the numerical aperture of the lens. We must, however, adopt the new coordinate system shown in Fig. 3(a) because now the z axis is not along one single ray, but represents the optical axis of the beam as a whole. For this situation,

**F**=

*F*

_{x}**x**̂+

*F*

_{z}**ẑ**.

*γ*is defined as the angle between the z axis and the distance vector

**r**=

*r*

**r**̂. We may regard the -z axis as the optical axis of the – focused – Gaussian beam, whose focus

*f*is the same as those in Figs. 1(a) and 1(b). A tridimensional perspective is given in Fig. 3(b), including the converging cone of the beam. Note that the apex angle is equivalent to 2

*θ*

_{max}, i.e., all incident rays are interior to or at the surface of this cone. Due to the symmetry of the Gaussian beam, the problem can be reduced to two dimensions or, explicitly, the

*x-z*plane (as mentioned before, the beam is circularly polarized and shifting the particle along

*x*or y makes no difference on the intensity of the total force

**F**, although its direction changes, as expected).

*F*⃗

^{i}=

*F*x̂+

^{i}_{x}*F*ẑ is the vector force of every incident ray in this new coordinate system and

^{i}_{z}*A*is the area of the sphere delimited by its intersection with the converging cone of the beam which is equivalent to consider one ray as being a differential

*dθdϕ*(inside the region 0≤

*θ*≤

*θ*

_{max}), and whose power is proportional to the lens’ surface area. The differential of this area is, using Abbe sine condition, proportional to sin

*θ*cos

*θdθdϕ, θ*and

*ϕ*associated with the usual spherical angles of a spherical coordinate system centered at the focus

*f*of the beam.

*θ*must be written as a function of

_{i}*γ*in order to evaluate Eq. (3). This can be accomplished by using the following relations, which can be deduced from Fig. 3(a) and simple geometrical considerations:

*γ*for |

**r**|=0.5

*a*and the same refractive index used before, still normalizing the forces over

*n*1

*P/c*, and perform the surface integral in Eq. (3) by making use of Eq. (4). The results for a conventional dielectric particle with

*n*

_{2}=1.62 is shown in Fig. 4(a), while the DNG case can be appreciated in Fig. 4(b). Physical interpretations for the first case are found elsewhere [14

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

*<*γ<135° and 235°

*<*γ<270°. Besides that, now the maximum values for the gradient force does not happen at

*γ*=90° and 270°, as in the conventional case, but for

*γ*≈120° and 240° instead. Finally, note the difference in magnitude for both plots, showing a more effective trapping for DNG particles or, in other words, that the same trapping effectiveness could be achieved for a DNG particle using less incident power.

*x*and

*z*-axis (see Fig. 4(c)), thus proving that particles where

*n*

_{2}>

*n*

_{1}are shifted toward regions of high intensity of the beam whereas, for

*n*

_{2}<

*n*

_{1}, towards regions of nulls of intensity, i.e., far away from the optical axis and the focus.

*n*

_{2}|<

*n*

_{1}, the total forces does not change sign, as shown in Fig. 4(d). Thus, and at least in theory, optical trapping for this kind of particle is unaffected when the relative refractive index changes from

*n*=|

*n*

_{2}|/

*n*

_{1}>1 to

*n*’=1/n<1.

### 2.2 Total forces as functions of r=|r|

*r*between the beam focus and the centre of the sphere for three different angles

*γ*=0°,

*γ*=90° and

*γ*=180°. Fig. 5 shows the forces for DNG particles.

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**, 156–159 (
1970). [CrossRef]

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

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

*γ*=0°. In this case, the scattering force is negative (scattering total forces will never be zero for

*r*=0), giving the vertical displacement of the particle towards the focus, located just below its center. We can see that this force diminishes not for

*r/a*≈1, but instead for

*r/a*>0.78, revealing a trapping efficiency diminishment. Yet, comparing the diminishment in magnitude with the conventional case, the total force is still much higher than that of a conventional particle. Therefore, in general, even in this case for 0.78<

*r/a*<1, the efficiency in trapping is higher when the particle presents negative refractive index (a better look in Fig. 4 could have already given this conclusion, as total gradient and scattering forces are higher in cases (b) and (d) than in (a) and (c), respectively).

*r/a*→1; however, this force is always negative, meaning that the particle will not have a point of stable equilibrium in the same horizontal plane of the focus; rather, this position will be slightly down. The gradient force reaches a maximum peak of ≈-0.9 arbitrary units (a.u.), more than twice that for a conventional particle when

*r/a*→1. A curious fact is the inversion of the gradient force when

*r/a*>0.88. This means that, for distances above

*r/a*=0.88, forces acting on this particle become repulsive, and no trapping is achieved.

*r/a*=0.73 – is, indeed, an unstable point. If we were capable of placing a metamaterial particle under

*γ*=180° and

*r/a*>0.73, the total force would be repulsive. For shorter distances between focus and centre of the sphere, it would become more attractive. It must be emphasized that this situation is quite hypothetical and, experimentally, placing a DNG particle in this unstable point is practically unrealizable.

*n*=

*n*

_{2}/

*n*

_{1}>1 and, to

*n*’=1/

*n*<1, to regions of lower intensities, i.e., the particle is repelled; for DNG particles, the trapping characteristics are conserved when we go from

*n*=|

*n*

_{2}|/

*n*

_{1}>1 to

*n*’=1/

*n*<1 (Figs. 5(d)–5(f)).

*Fundação de Amparo à Pesquisa e ao Ensino do Estado de São Paulo*, under contracts 2005/54265-9 (L. A. Ambrosio’s Ph D grant) and 2005/51689-2 (CePOF, Optics and Photonics Research Center).

## References and links

1. | A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. |

2. | A. Ashkin, “Atomic-beam deflection by resonance-radiation pressure,” Phys. Rev. Lett. |

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

4. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

5. | A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science |

6. | 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 |

7. | 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,” |

8. | V. Emiliani |

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. | J. B. Pendry, “Negative Refraction Makes Perfect Lens,” Phys. Rev. Lett. |

12. | N. Engheta and R. W. Ziolkowski, |

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

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

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(160.3918) Materials : Metamaterials

**ToC Category:**

Optical Tweezers or Optical Manipulation

**History**

Original Manuscript: September 14, 2009

Revised Manuscript: October 27, 2009

Manuscript Accepted: November 5, 2009

Published: November 16, 2009

**Citation**

Leonardo 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**, 21918-21924 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21918

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### References

- A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970). [CrossRef]
- A. Ashkin, "Atomic-beam deflection by resonance-radiation pressure," Phys. Rev. Lett. 24, 1321-1324 (1970). [CrossRef]
- A. Ashkin and J. M. Dziedzic "Optical levitation by radiation pressure," Appl. Phys. Lett. 19, 283-285 (1971). [CrossRef]
- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986). [CrossRef] [PubMed]
- A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 235, 1517-1520 (1987). [CrossRef] [PubMed]
- 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, 505-510 (1991). [CrossRef] [PubMed]
- 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, 7914-7918 (1989). [CrossRef]
- V. Emiliani et al., "Wave front engineering for microscopy of living cells," Opt. Express 13, 1395-1405 (2005). [CrossRef] [PubMed]
- V. G. Veselago, "The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- 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 (2000). [CrossRef]
- J. B. Pendry, "Negative Refraction Makes Perfect Lens," Phys. Rev. Lett. 85, 18 (2000). [CrossRef]
- N. Engheta and R. W. Ziolkowski, Metamaterials - Physics and Engineering Explorations, (IEEE Press, Wiley-Interscience, 2006).
- N. Engheta and R. W. Ziolkowski, "A positive future for Double-negative metamaterials," IEEE Trans. Microwave Theory Tech. 53(4), (part II), 1535-1556 (2005).
- A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime," Biophys. J. 61, 569-582 (1992). [CrossRef] [PubMed]

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