## Tailored optical force fields using evolutionary algorithms |

Optics Express, Vol. 19, Issue 19, pp. 18543-18557 (2011)

http://dx.doi.org/10.1364/OE.19.018543

Acrobat PDF (9654 KB)

### Abstract

We introduce a method whereby the electromagnetic field that governs the force on a Rayleigh particle can be tailored such that the resultant force field conforms to a desired geometry. The electromagnetic field is expanded as a set of vector spherical wavefunctions (VSWFs) that describe the field over all space. Given the incident field, the resultant force on a given Rayleigh particle can be calculated throughout a volume of interest. We use an evolutionary algorithm (EA) to search the space of coefficients governing the VSWFs for those that produce the desired force field. We demonstrate how Maxwell’s equations will support an “optical tunnel” that guides particles to a trap location while at the same time preventing particles outside the tunnel from approaching the trap. This result is of interest because the field is impressed throughout the domain; that is to say, once the field is generated, no additional control is required to guide the particles.

© 2011 OSA

## 1. Introduction

1. 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]

2. D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. **15**, 041503 (2010). [CrossRef] [PubMed]

4. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–816 (2003). [CrossRef] [PubMed]

5. D. L. Andrews, *Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces* (Elsevier, 2008). [PubMed]

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

12. J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express **12**, 220–226 (2004). [CrossRef] [PubMed]

13. T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B **43**, 102001 (2010). [CrossRef]

14. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. **21**, 827–829 (1996). [CrossRef] [PubMed]

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

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

17. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. **25**, 191–193 (2000). [CrossRef]

18. A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. **133**, 7–10 (1997). [CrossRef]

19. P. C. Morgensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. **175**, 75–81 (2000). [CrossRef]

20. E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. **69**, 1974–1977 (1998). [CrossRef]

21. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. **24**, 608–610 (1999). [CrossRef]

22. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. **60**, 807–809 (1992). [CrossRef]

26. R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A **13**, 1837–1848 (1996). [CrossRef]

29. G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. **7**, 117 (2005). [CrossRef]

## 2. Field expansion

**E**(

**r**) satisfies with

*k*=

*n*/

_{m}ω*c*,

*n*the refractive index of the medium,

_{m}*ω*the optical angular frequency, and

*c*the speed of light in vacuum. We define a polar spherical coordinate system centered at the Cartesian coordinates

*x*= 0,

*y*= 0,

*z*= 0 with position vector

**r**= (

*x*,

*y*,

*z*) = (

*r*,

*θ*,

*ϕ*) and respective unit vectors

*x ̂ŷ*,

*ẑ*and

*r̂*,

*,*θ ^

*. The co-latitude,*ϕ ^

*θ*, is measured from the +

*z*-axis and the azimuth,

*ϕ*, is measured from the +

*x*-axis toward the +

*y*-axis. The VSWFs are a complete basis set of solutions to Eq. (2), thus, any arbitrary complex amplitude can be written as a linear combination of the VSWFs as follows: where

*A*is an overall field amplitude coefficient with dimensions of electric field and the expansion coefficients

*ε*

_{0}and

*μ*

_{0}are the permittivity and permeability of free space, respectively, and

*ε*and

_{r}*μ*are the dimensionless relative permittivity and permeability of the medium [30

_{r}30. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B **22**, 1620–1631 (2005). [CrossRef]

*j*(

_{n}*kr*) is a spherical Bessel function and

30. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B **22**, 1620–1631 (2005). [CrossRef]

33. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. **79–80**, 1005–1017 (2003). [CrossRef]

*N*as the upper limit of the summation in Eq. (3) we can determine

_{max}**E**(

**r**) at any point within the volume of interest given values for the complex expansion coefficients

**E**(

**r**) as discussed in [33

33. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. **79–80**, 1005–1017 (2003). [CrossRef]

**E**(

**r**) given desired particle trajectories in the volume of interest.

## 3. Force field calculation

34. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. **124**, 529–541 (1996). [CrossRef]

*n*is the index of the particle and

_{p}*R*is its radius. Using the relation,

**E**(

**r**)|

^{2}, can be calculated numerically using finite differences if

**E**(

**r**) is known. The Poynting vector represents the instantaneous energy flux crossing a unit area per unit time and is given as with the time-dependent magnetic field given by The scattering force on a particle is given by where is the time-average of the Poynting vector and is the scattering cross-section of the particle in the Rayleigh regime. Finally, the total force on the particle is simply Thus, the force that a Rayleigh particle will encounter anywhere in a beam defined by a given set of expansion coefficients,

*ã*, can be determined by numerically calculating the gradient field and using Eqs. (3), (4), (12), (15), (16), and (18).

*A*in Eq. (3) is required to appropriately account for beam power. There has been some disagreement in the literature about how incident power,

*P*, should be calculated for beams expanded as VSWFs. Moine and Stout [30

_{i}30. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B **22**, 1620–1631 (2005). [CrossRef]

*P*∝ ||

_{i}*ã*||

^{2}, where the double-bars represent the Euclidean norm. This proportionality was later shown by Simpson and Hanna [35

35. S. H. Simpson and S. Hanna, “Rotation of absorbing spheres in Laguerre–Gaussian beams,” J. Opt. Soc. Am. A **26**, 173–183 (2009). [CrossRef]

*A*in Eq. (21) of [35

35. S. H. Simpson and S. Hanna, “Rotation of absorbing spheres in Laguerre–Gaussian beams,” J. Opt. Soc. Am. A **26**, 173–183 (2009). [CrossRef]

*μ*=

_{m}*μ*

_{r}μ_{0}and

*ε*=

_{m}*ε*

_{r}ε_{0}are the permeability and permittivity of the medium.

34. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. **124**, 529–541 (1996). [CrossRef]

**22**, 1620–1631 (2005). [CrossRef]

*m*≠ |1| are zero,

*ê*is a polarization vector with ||

_{i}*ê*|| = 1, and

_{i}*ã*has been replaced by

*g*are beam-shape coefficients [36

_{n}36. G. Gouesebet, J. A. Locke, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. **34**, 2133–2143 (1995). [CrossRef]

37. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A **19**, 1177–1179 (1979). [CrossRef]

*s*

^{2}power series approximations of the vector potential, where the dimensionless beam shape parameter is defined by and

*w*

_{0}is the beam waist radius in the paraxial approximation. The Davis model is useful here because it allows the

*g*to be calculated analytically by either a first-, third-, or fifth-order expansion given, respectively, as

_{n}*ã*for a focused Gaussian beam and are now in a position to compare

**F**

*(*

_{grad}**r**) and

**F**

*(*

_{scat}**r**) calculated using (12) and (15) to analytical representations of the same.

34. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. **124**, 529–541 (1996). [CrossRef]

*ẑ*-direction because the model breaks down with highly focused beams and the assumption of only one transverse component of the electric field is accurate given the value of

*s*= 0.012 used in their work. The analytical descriptions of the gradient forces in the

*x̂*-,

*ŷ*-, and

*ẑ*-directions are given, respectively, by

**124**, 529–541 (1996). [CrossRef]

*ê*= [1 0]) beam acting on polystyrene spheres in water was assumed in [34

_{i}**124**, 529–541 (1996). [CrossRef]

*n*= 1.592,

_{p}*n*= 1.332,

_{m}*w*

_{0}= 5

*μ*m,

*P*= 100 mW, and a vacuum wavelength

_{i}*λ*

_{0}= 0.5145

*μ*m. We use the same values and calculate the Gaussian fields using

*N*= 130. Analytical and computational results for the Gaussian beam are compared in Figure 1.

_{max}**F**

*(*

_{T}**r**), by generating a Gaussian beam for a particle with

*R*= 50 nm. Fewer coefficients (

*N*= 12) are required to accurately represent a more focused beam with

_{max}*w*

_{0}=

*λ*where

_{m}*λ*=

_{m}*λ*

_{0}/

*n*. We discretize the

_{m}*z*= 0 plane using increments of 0.2

*λ*in the

_{m}*x*- and

*y*-directions and calculate the force field corresponding to a particle with the same properties used above to yield

**F**

*(*

_{g}**r**)|

_{z}_{=0}. This field is then modified by leaving the central core of the field unchanged and inverting the sign of forces in the plane outside the central core as shown in Figure 2A. The target field in the

*z*= 0 plane is then copied at additional

*z*-planes to build the full 3D optical tunnel as shown in Figure 2B. Care must be taken to ensure that the spacing between sampled

*z*planes is small enough to accurately capture all relevant spatial structure of the beam. Here, a spacing of 0.1

*λ*between planes ensures that the computed gradient in the

_{m}*z*-direction is accurate.

## 4. Design procedure

**F**

*(*

_{T}**r**) in a least-squares sense. We combine all the coefficients into a single vector by letting and defining If we seek to match all force components comprising

**F**

*(*

_{T}**r**) over the sampled volume of interest, then we are solving an optimization problem given by where

**R**is the set of all discretized points,

**r**

*, that we have sampled in the volume of interest and*

_{n}**F**

*(*

_{t}**r**) is a test force field associated with a given set of expansion coefficients,

**v**

*. Designating*

_{t}*𝒜*as the number of coefficients corresponding to a given

*N*, the dimension of the search space for the optimization (31) is 2

_{max}*𝒜*which can become quite large in the fully general case of

*m*| = 1 are required and the number of coefficients scales as

*𝒜*= 4

*N*[33

_{max}33. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. **79–80**, 1005–1017 (2003). [CrossRef]

**v**would need to be estimated numerically. For any reasonable value of

*N*this would quickly become prohibitive. Furthermore, such an algorithm is highly likely to converge to a local optimum if the search space is composed of multiple optima.

_{max}*a priori*that the search space is convex, we make use of a global direct search routine to search the parameter space, in particular, a specific type of evolutionary algorithm called differential evolution (DE) [38

38. R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. **11**, 341–359 (1997). [CrossRef]

**v**, governing

**E**(

**r**). During each generation, the algorithm steps through each member of the population and compares the fitness of that member (the “target vector”) to the fitness of a new vector that is created from the population (the “trial vector”). Each trial vector is created in the following manner: first, a “difference vector” is created by taking the difference between two randomly selected members of the population and multiplying by a scale factor

*F*. Next, a third member of the population is randomly selected and added to the scaled difference vector to yield a composite vector. The final trial vector is created by populating each element of the vector with an element from either the target vector or the composite vector where an element from the target vector is selected with probability

_{s}*CR*. Subsequently, the fitness of the trial vector is calculated and compared to the fitness of the target vector with the higher-fitness vector selected as a member of the next generation population. In this work we use

*F*= 0.9 and

_{s}*CR*= 0.5 as recommended in [38

38. R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. **11**, 341–359 (1997). [CrossRef]

## 5. Results and discussion

*z*planes. All of the runs converged to similar solutions.

*z*/

*λ*= −1. Each particle was always assumed to instantaneously move at terminal velocity within the fluid (assuming Stokes drag) and a finely sampled cube of points centered at the particle’s current location was used to determine the local forces present on the particle. The results of this exercise are shown in Figures 4 and 5.

*z*/

*λ*= 1 plane) where particles outside the core region circle about the center. The algorithm seeks to maximize the objective function over the optimization region but can only do so in a least-squares sense under the constraints of Maxwell’s equations. A more tunnel-like field might be possible if the size of the particle were increased such that the scattering force contributed more to the overall field. In addition, a longer tunnel region could be generated if the target field were extended, but at the expense of increased computation time. Regardless, the present field does succeed in generating a core region that is protected from particles that originate beyond the core boundary.

*z*-component of the Poynting vector at various points along the

*z*-axis as well as all components of the Poynting vector at the

*z*= 0 plane. The structure of the discovered beam is quite different from both Laguerre-Gauss [14

14. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. **21**, 827–829 (1996). [CrossRef] [PubMed]

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

*z*-axis. In particular, the relative strength of the high-intensity rings changes significantly, which is not typical of Bessel or Laguerre-Gauss beams. The direction of the Poynting vector also oscillates at points located on the

*z*-axis, which is another differentiating feature of the discovered beam.

39. R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. **19**, 771–773 (1994). [CrossRef] [PubMed]

*a priori*whether the desired field is consistent with Maxwell’s equations [26

26. R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A **13**, 1837–1848 (1996). [CrossRef]

41. G. C. Spalding, J. Courtial, and R. Di Leonardo “Holographic Optical Tweezers,” in *Structured Light and Its Applications*, D. L. Andrews, ed. (Elsevier, 2008), Chap. 6. [CrossRef]

29. G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. **7**, 117 (2005). [CrossRef]

42. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. **26**, 2788–2798 (1987). [CrossRef] [PubMed]

43. C. H. Choi, J. Ivanic, M. S. Gordon, and K. Ruedenberg, “Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion,” J. Chem. Phys. **111**, 8825–8831 (1999). [CrossRef]

46. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A **9**, S196–S203 (2007). [CrossRef]

*a*≈

*λ*could be determined throughout the optimization volume. In addition, the T-matrix framework can be naturally extended to non-spherical particles.

27. R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. **43**, 1495–1507 (1996). [CrossRef]

## 6. Conclusion

## References and links

1. | 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. |

2. | D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. |

3. | A. Jonáš and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis |

4. | D. G. Grier, “A revolution in optical manipulation,” Nature |

5. | D. L. Andrews, |

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

7. | T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” App. Phys. Lett. |

8. | V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” App. Phys. Lett. |

9. | J. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. |

10. | J. B. Wills, J. R. Butler, J. Palmer, and J. P. Reid, “Using optical landscapes to control, direct, and isolate aerosol particles,” Phys. Chem. Chem. Phys. |

11. | J. Liesner, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. |

12. | J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express |

13. | T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B |

14. | K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. |

15. | J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. |

16. | V. Garcés-Chávez, D. McGloin, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

17. | J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. |

18. | A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. |

19. | P. C. Morgensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. |

20. | E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. |

21. | M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. |

22. | K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. |

23. | C. Mio, T. Gong, A. Terray, and D. W. M. Marr, “Design of a scanning laser optical trap for multiparticle manipulation,” Rev. Sci. Instrum. |

24. | K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. |

25. | M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. |

26. | R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A |

27. | R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. |

28. | G. Shabtay, “Three-dimensional beam forming and Ewald’s surfaces,” Opt. Commun. |

29. | G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. |

30. | O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B |

31. | J. D. Jackson, |

32. | L. Tsang, J. A. Kong, and R. T. Shin, |

33. | T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. |

34. | Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. |

35. | S. H. Simpson and S. Hanna, “Rotation of absorbing spheres in Laguerre–Gaussian beams,” J. Opt. Soc. Am. A |

36. | G. Gouesebet, J. A. Locke, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. |

37. | L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A |

38. | R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. |

39. | R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. |

40. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik |

41. | G. C. Spalding, J. Courtial, and R. Di Leonardo “Holographic Optical Tweezers,” in |

42. | M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. |

43. | C. H. Choi, J. Ivanic, M. S. Gordon, and K. Ruedenberg, “Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion,” J. Chem. Phys. |

44. | G. Videen, “Light Scattering from a Sphere Near a Plane Surface,” in |

45. | T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. |

46. | T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A |

47. | B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(140.7010) Lasers and laser optics : Laser trapping

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

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: May 23, 2011

Revised Manuscript: July 19, 2011

Manuscript Accepted: July 19, 2011

Published: September 8, 2011

**Virtual Issues**

Vol. 6, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Colin C. Olson, Ross T. Schermer, and Frank Bucholtz, "Tailored optical force fields using evolutionary algorithms," Opt. Express **19**, 18543-18557 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18543

Sort: Year | Journal | Reset

### References

- 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]
- D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light forces the pace: optical manipulation for biophotonics,” J. Biomed. Opt. 15, 041503 (2010). [CrossRef] [PubMed]
- A. Jonáš and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813–4851 (2008). [CrossRef]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]
- D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, 2008). [PubMed]
- P. J. Rodrigo, V. R. Daria, and J. Glückstad, “Four-dimensional optical manipulation of colloidal particles,” App. Phys. Lett. 86, 074103 (2005). [CrossRef]
- T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” App. Phys. Lett. 86, 174101 (2005). [CrossRef]
- V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” App. Phys. Lett. 86, 031106 (2005). [CrossRef]
- J. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002). [CrossRef]
- J. B. Wills, J. R. Butler, J. Palmer, and J. P. Reid, “Using optical landscapes to control, direct, and isolate aerosol particles,” Phys. Chem. Chem. Phys. 11, 8015–8020 (2009). [CrossRef] [PubMed]
- J. Liesner, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185, 77–82 (2000). [CrossRef]
- J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express 12, 220–226 (2004). [CrossRef] [PubMed]
- T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010). [CrossRef]
- K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef] [PubMed]
- J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001). [CrossRef]
- V. Garcés-Chávez, D. McGloin, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002). [CrossRef] [PubMed]
- J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000). [CrossRef]
- A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997). [CrossRef]
- P. C. Morgensen and J. Glückstad, “Dynamic array generation and pattern formation for optical tweezers,” Opt. Commun. 175, 75–81 (2000). [CrossRef]
- E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998). [CrossRef]
- M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999). [CrossRef]
- K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal-particle and a water droplet by a scanning laser-beam,” App. Phys. Lett. 60, 807–809 (1992). [CrossRef]
- C. Mio, T. Gong, A. Terray, and D. W. M. Marr, “Design of a scanning laser optical trap for multiparticle manipulation,” Rev. Sci. Instrum. 71, 2196–2200 (2000). [CrossRef]
- K. Visscher, S. P. Gross, and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1075 (1996). [CrossRef]
- M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Commun. 33, 599–601 (2008).
- R. Piestun, B. Spektor, and J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996). [CrossRef]
- R. Piestun, B. Spektor, and J. Shamir, “Unconventional light distributions in three-dimensional domains,” J. Mod. Opt. 43, 1495–1507 (1996). [CrossRef]
- G. Shabtay, “Three-dimensional beam forming and Ewald’s surfaces,” Opt. Commun. 226, 33–37 (2003). [CrossRef]
- G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J. Phys. 7, 117 (2005). [CrossRef]
- O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B 22, 1620–1631 (2005). [CrossRef]
- J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
- L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
- T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003). [CrossRef]
- Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996). [CrossRef]
- S. H. Simpson and S. Hanna, “Rotation of absorbing spheres in Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 26, 173–183 (2009). [CrossRef]
- G. Gouesebet, J. A. Locke, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995). [CrossRef]
- L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979). [CrossRef]
- R. Storn and R. Price, “Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim. 11, 341–359 (1997). [CrossRef]
- R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994). [CrossRef] [PubMed]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- G. C. Spalding, J. Courtial, and R. Di Leonardo “Holographic Optical Tweezers,” in Structured Light and Its Applications , D. L. Andrews, ed. (Elsevier, 2008), Chap. 6. [CrossRef]
- M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987). [CrossRef] [PubMed]
- C. H. Choi, J. Ivanic, M. S. Gordon, and K. Ruedenberg, “Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion,” J. Chem. Phys. 111, 8825–8831 (1999). [CrossRef]
- G. Videen, “Light Scattering from a Sphere Near a Plane Surface,” in Light Scattering from Microstructures, Lecture Notes in Physics Volume 534 , F. Moreno and F. González, eds. (Springer, 2000), Chapter 5. [CrossRef]
- T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003). [CrossRef]
- T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007). [CrossRef]
- B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765–15776 (2008). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.