## Three-dimensional forces in GPC-based counterpropagating-beam traps

Optics Express, Vol. 14, Issue 12, pp. 5812-5822 (2006)

http://dx.doi.org/10.1364/OE.14.005812

Acrobat PDF (848 KB)

### Abstract

We theoretically investigate the three-dimensional (3D) trapping force acting on a microsphere held in a pair of counterpropagating beams produced by the generalized phase contrast (GPC) method. In the case of opposing beams of equal power, we identify the range of beam waist separation *s* that results in a stable 3D optical potential-well by assessing the dependence of the axial and transverse force curves on *s*. We also examine how the force curves are influenced by other parameters such as size and refractive index of the microsphere. Aside from force curves of beam tandems with equal powers, we also numerically calculate force curves for cases of beam pairs having disparate relative strengths. These calculations enable us to elucidate the large dynamic range for axial position control of microparticles in GPC-based counterpropagting-beam traps.

© 2006 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. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. **18**, 1867–1869 (1993). [CrossRef] [PubMed]

3. E. R. Lyons and G. J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. **66**, 1584–1586 (1995). [CrossRef]

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

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

*s*

_{c}results in an unstable potential-well with possible emergence of additional multiple points of stable equilibrium. Furthermore, one can deduce in Ref. 5 that the value of

*s*

_{c}increases with microsphere diameter. Trap efficiencies were also investigated as functions of parameters such as the microsphere’s refractive index and radius. However, only the case of equally powerful opposing beams was analyzed.

6. P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. **29**, 2270–2272 (2004). [CrossRef] [PubMed]

9. P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express **13**, 6899–6904 (2005). [CrossRef] [PubMed]

*s*

_{c}corresponding to a symmetric GPC-based CB trap to find the operational beam waist separation and compare with our previous experimental implementations. GPC-based CB traps are real-time and interactively reconfigurable arrays of optical traps that rely on the energy-efficient intensity-visualization of phase patterns imparted on an incident plane wave by a spatial light modulator [10

10. J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. **130**, 225–230 (1996). [CrossRef]

11. J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. **40**, 268–282 (2001). [CrossRef]

6. P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. **29**, 2270–2272 (2004). [CrossRef] [PubMed]

9. P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express **13**, 6899–6904 (2005). [CrossRef] [PubMed]

*x*,

*y*, and

*z*-components of the trapping force acting on a sphere due to a GPC-based CB trap. In section 3, we obtain the intensity distribution describing the propagation of one of the constituent beams through the host medium. In section 4, the analysis in sections 2 and 3 are used to numerically calculate the axial and transverse force curves as functions of beam waist separation, particle refractive index, and particle radius. We also propose a force constant (trap stiffness) versus

*s*plot as a quantitative means of finding a critical separtation

*s*

_{c}. Finally, the axial force curves for counterpropagating beams with unequal strengths are used to estimate the CB trap’s dynamic range of axial position control.

## 2. Three-dimensional trapping force components

*a*illuminated by two coaxial (i.e. along

*z*-axis) counterpropagating orthogonally polarized optical fields (Fig. 1(a)) corresponding to intensity distributions described by

*I*

^{+}(

*x*,

*y*,

*z*) and

*I*

^{-}(

*x*,

*y*,

*z*) where +(-) denotes a beam propagation directed towards the positive (negative)

*z*-axis. First, we examine the forces induced by

*I*

^{+}(

*x*,

*y*,

*z*) on the particle. Guided by Fig. 1(b), two force components, namely the parallel force d

**F**^{+}

_{‖}and the perpendicular force d

**F**^{+}

_{⊥}, may be resolved when a ray is incident to an infinitesimal region d

*S*on the sphere’s surface:

*R*and transmittance

*T*are carefully determined by noting the polarization of the incident field and the plane of incidence. The differential power d

*P*through d

*S*is given by

*x*

_{p},

*y*

_{p},

*z*

_{p}) refer to the point of incidence P with respect to the unprimed rectangular coordinate system in Fig. 2. The effective forces acting along three Cartesian axes can be obtained from the integration of the scalar products of d

**F**^{+}

_{‖}d

**F**^{+}

_{⊥}with unit vectors

*x̂*,

*ŷ*, and

*ẑ*over the illuminated sphere surface.

*ê*

_{‖}=

*ẑ*and

*ê*

_{⊥}=

*x̂*cos

*φ*+

*ŷ*sin

*φ*and that

*α*

_{i}=

*θ*(of the spherical coordinates of point P with respect to sphere center O positioned at (

*x*

_{o},

*y*

_{o},

*z*

_{o})), we obtain

*x*

_{p}=

*x*

_{o}+

*a*sin

*θ*cos

*φ*,

*y*

_{p}=

*y*

_{o}+

*a*sin

*θ*sin

*φ*, and

*z*

_{p}=

*z*

_{o}-

*a*cos

*θ*are used to determine the intensity at different points on the surface of integration. The Fresnel integral [12] is used to numerically obtain the intensity

*I*

^{+}(

*x*

_{p},

*y*

_{p},

*z*

_{p}), which is a result of the free-medium propagation of the function

*e*

^{+}(

*x*,

*y*) describing the field distribution at the objective focal plane

*z*=0.

*x*-axis. Since different planes of incidence exist for various points on the illuminated sphere surface, each of the force components is calculated for the TE and the TM components separately, and afterward the contributions are combined. Therefore, in addition to the proper calculation of

*R*and

*T*in Eq. (2a) and Eq. (2b), one must also multiply the integrands of Eqs. (4) a factor of sin

^{2}φ or cos

^{2}

*φ*that corresponds to the fraction of the input power carried by the TE or TM component.

## 3. Tophat field distribution and propagation

*z*=0 when forming a GPC-based counterpropagating-beam trap may be approximated by a tophat profile

*P*

^{+}is uniformly distributed over the circle of radius

*R*. The 3D intensity distribution

*I*

^{+}(

*x*,

*y*,

*z*) for

*z*>0 is obtained by numerically calculating the Fresnel integral [12]. This corresponds to the diffraction pattern that the field distribution described by Eq. (5) undergoes upon propagation within the medium of refractive index

*n*

_{1}.

*xz*-plane. In this sample calculation we used

*R*=1.5 µm and

*P*

^{+}=5 mW, which are typical values realized in our past experiments. An observation one can make from Fig. 3 is that a global intensity maximum (about four times the intensity

*P*

^{+}/

*πR*

^{2}at

*z*=0) occurs at

*z*~3.8 µm which is consistent with the predicted position

*z*

_{I,max}=

*n*

_{1}

*R*

^{2}/

*λ*, when considering that the beam (

*λ*=830 nm) propagates in water - commonly used particle host medium with index

*n*

_{1}=1.33. It is also worth to note that a plane defined by

*z*=

*z*

_{I,max}separates the Fresnel and the Fraunhofer regions of the diffraction pattern. In the Fraunhofer region, one finds a central lobe that decreases in peak intensity but increases in width for increasing

*z*position. In the section that follows, we use these observations to explain the obtained dependence of axial and transverse forces on the position of the particle within the CB trap.

## 4. Numerical calculation of force curves

*P*

_{t}=

*P*

^{+}+

*P*

^{-}, we instead calculate for the trapping parameters given by

*P*

^{-}directed towards negative z-axis (see Fig. 1(a)). Numerical calculations of

*P*

^{+}in Eq. (5) is replaced by

*P*

^{-}and

*z*

_{o}, which denotes the axial position of the microsphere, is replaced by

*s*-

*z*

_{o}. As depicted in Fig. 1(a), the beam waist separation

*s*is defined here as the distance separating the two microscope objective focal planes where the two GPC-generated tophat beams are imaged.

*n*

_{2}=1.59 ; radius

*a*=1.5 µm) in water, we first look at the case where the opposing tophat beams are identical in power,

*P*

^{+}=

*P*

^{-}, and in size

*R*=1.5 µm. Figures 4(a) and 4(b) illustrate, for different waist separation

*s*, the axial and transverse force profiles, respectively - axial (transverse) force as a function of axial

*z*

_{o}-

*s*/2 (transverse

*x*

_{o}) offset. Note that the force curves feature an odd-function property indicating a trapping potential with both axial and transverse symmetry about the separation midpoint. The plots in Fig. 4(a) also clearly show that, except for

*s*=20 µm, a GPC-based CB trap provides a stable harmonic axial potential-well as characterized by the linear restoring force (i.e. negative slope) about the equilibrium position. For

*s*=20 µm however, one obtains at the separation midpoint a positive slope in the axial force curve making it an axially unstable point of equilibrium. Although at the midpoint (in this case given by (0,0,

*s*/2=10 µm) of the

*xyz*-coordinate system) the trap potential appears to be transversely stable and harmonic (Fig. 4(b)), a small perturbation in the particle’s axial position will cause it to be ejected into either direction of the

*z*-axis. These results are reminiscent of those for a fiber-based CB trap with a Gaussian mode profile [5

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

*f*

_{x}and

*f*

_{z}on

*s*as shown in Fig. 5. Here we assume that, for radial distances from the midpoint (0,0,

*s*2) comparable with the particle radius,

*Q*

_{xt}≈-

*f*

_{x}

*x*

_{o}and

*Q*

_{zt}≈-

*f*

_{z}(

*z*

_{o}-

*s*/2). For the pertinent range of

*s*considered, our findings indicate that

*f*

_{x}decreases with

*s*but remains positive-valued. This decay of transverse trap stiffness with

*s*can be easily understood by recognizing the intensity profiles illuminating the microsphere near the trapping point. One finds that larger

*s*would translate to smaller intensity gradient along the transverse direction as the two counterpropagating beams illuminate the particle with broader central lobe in their respective Fraunhofer regions (see Fig. 3). Note also that the same reason explains the slight increase in the transverse attractive range with

*s*(see Fig. 4(b)). On the contrary,

*f*

_{z}can either be positive giving a stable trap when

*s*>

*s*

_{c}or negative forming an unstable trap, when

*s*<

*s*

_{c}(i.e.

*f*

_{z}(

*s*

_{c})=0). Clearly,

*s*

_{c}may be accurately obtained as the zero-crossing of a

*f*

_{z}(

*s*)-curve as exemplified by Fig. 5. Furthermore, the use of

*f*

_{z}(

*s*)-curve to identify a maximum axial force constant

*f*

_{z,max}also suggests a procedure for choosing an optimal beam waist separation

*s*

_{op}, for example, by defining

*f*

_{z}(

*s*

_{op})=

*f*

_{z,max}. In Fig. 5, we find

*s*

_{op}=30 µm which matches the estimated value applied in our previous experiments with the same set of parameters [6

6. P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. **29**, 2270–2272 (2004). [CrossRef] [PubMed]

7. P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. **86**, 074103 (2005). [CrossRef]

*s*

_{op}may also take into account the decrease of

*f*

_{x}with

*s*and other parameters like size of the largest particle (for size-varied colloids) and desired dynamic axial range of manipulation as discussed below.

*s*>

*s*

_{c}, Figs. 6(a) and 6(b) illustrate the axial and transverse force curves for different particle indices

*n*

_{2}. The result simply confirms that both force components increase with refractive index contrast between particle and suspending medium due to the expected increase in momentum change that light undergoes at the

*n*

_{1}–

*n*

_{2}interface. Meanwhile, Figs. 7(a) and 7(b) provide the force curves for different sphere radii

*a*. For sphere radii

*a*≤

*R*, a smaller sphere captures lesser beam power and thus translate to weaker forces particularly near the equilibrium point and thus giving weaker trap stiffness. In addition, the location of the leading axial force maximum is closer to the point of stable equilibrium for larger

*a*as shown in Fig. 7(a). This is expected to diminish the axial range within which the particle can be stably positioned.

*P*

^{+}-

*P*

^{-}. In our simulation, we pattern the way

*P*

^{+}-

*P*

^{-}is changed on how it is being efficiently done experimentally with a polarization scheme [6

**29**, 2270–2272 (2004). [CrossRef] [PubMed]

9. P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express **13**, 6899–6904 (2005). [CrossRef] [PubMed]

*P*

_{t}=

*P*

^{+}+

*P*

^{-}. From zero differential power, we tune

*P*

^{+}-

*P*

^{-}to a value that shifts the stable point of equilibrium to a cutoff axial offset Δ

*z*where the axial force constant decreases by 50%. The axial offset from -Δ

*z*to +Δ

*z*defines a competitively large dynamic range for axial position control of the particle. We also plot the axial force curve that gives another asymmetric potential with stable equilibrium point found at an axial offset of Δ

*z*/2. At

*xy*-planes containing the axially displaced equilibrium points, the transverse force curves are determined. The manner in which the associated trap stiffness vary with the equilibrium position are shown in the respective insets.

*f*

_{z}(

*s*)-curves for microsphere radii

*a*=1.0 µm, 1.5 µm, and 2.5 µm, say, to pre-analyze a trapping experiment with a size-polydisperse sample. The plot is indicative of the role that particle radius

*a*plays in identifying an operational beam waist separation

*s*

_{op}>

*s*

_{c}. From the figure, one may anticipate the relative trap stiffness for the two smaller microspheres and the inability to (axially) trap the largest if the chosen

*s*

_{op}=30 µm (where

*f*

_{z,max}for microsphere with

*a*=1.5 µm occurs). To simultaneously trap the three particles using three identical, symmetric CB traps having sufficient power, Fig. 9 suggest that

*s*

_{op}>35 µm must be implemented.

*s*

_{op}= 40 µm, which results only in a stable trap potential as shown by the force curves in Fig. 7 for the three different particle radii.

## 5. Conclusion

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

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

*a*≫

*λ*, the geometrical optics approximation may offer accurate results. However, good agreement of results from geometrical optics approach with those obtained experimentally has also been achieved for particle sizes comparable to the wavelength [3

3. E. R. Lyons and G. J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. **66**, 1584–1586 (1995). [CrossRef]

*a*≪

*λ*, calculation of the optical forces entails electromagnetic wave theory.

## Acknowledgments

## References and links

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

2. | A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. |

3. | E. R. Lyons and G. J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. |

4. | G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A |

5. | E. Sidick, S. D. Collins, and A. Knoesen, “Trapping forces in a multiple-beam fiber-optic trap,” Appl. Opt. |

6. | P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Real-time three-dimensional optical micromanipulation of multiple particles and living cells,” Opt. Lett. |

7. | P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” Appl. Phys. Lett. |

8. | I. R. Perch-Nielsen, P. J. Rodrigo, and J. Glückstad, “Real-time interactive 3D manipulation of particles viewed in two orthogonal observation planes,” Opt. Express |

9. | P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express |

10. | J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. |

11. | J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. |

12. | J. W. Goodman, |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

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

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Trapping

**History**

Original Manuscript: January 30, 2006

Revised Manuscript: May 30, 2006

Manuscript Accepted: May 30, 2006

Published: June 12, 2006

**Virtual Issues**

Vol. 1, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Peter John Rodrigo, Ivan R. Perch-Nielsen, and Jesper Glückstad, "Three-dimensional forces in GPC-based counterpropagating-beam traps," Opt. Express **14**, 5812-5822 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-12-5812

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### References

- A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970). [CrossRef]
- A. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, "Demonstration of a fiber-optical light-force trap," Opt. Lett. 18, 1867-1869 (1993). [CrossRef] [PubMed]
- E. R. Lyons and G. J. Sonek, "Confinement and bistability in a tapered hemispherically lensed optical fiber trap," Appl. Phys. Lett. 66, 1584-1586 (1995). [CrossRef]
- G. Roosen and C. Imbert, "Optical levitation by means of two horizontal laser beams: a theoretical and experimental study," Phys. Lett. A 59, 6-8 (1976). [CrossRef]
- E. Sidick, S. D. Collins, and A. Knoesen, "Trapping forces in a multiple-beam fiber-optic trap," Appl. Opt. 36, 6423-6433 (1997). [CrossRef]
- P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Real-time three-dimensional optical micromanipulation of multiple particles and living cells," Opt. Lett. 29, 2270-2272 (2004). [CrossRef] [PubMed]
- P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Four-dimensional optical manipulation of colloidal particles," Appl. Phys. Lett. 86, 074103 (2005). [CrossRef]
- I. R. Perch-Nielsen, P. J. Rodrigo, and J. Glückstad, "Real-time interactive 3D manipulation of particles viewed in two orthogonal observation planes," Opt. Express 18,2852-2857 (2005). [CrossRef]
- P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, "Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps," Opt. Express 13,6899-6904 (2005). [CrossRef] [PubMed]
- J. Glückstad, "Phase contrast image synthesis," Opt. Commun. 130, 225-230 (1996). [CrossRef]
- J. Glückstad and P. C. Mogensen, "Optimal phase contrast in common-path interferometry," Appl. Opt. 40, 268-282 (2001). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, Second Edition (McGraw-Hill, New York, 1996).

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