## FDTD simulations of forces on particles during holographic assembly

Optics Express, Vol. 16, Issue 5, pp. 2942-2957 (2008)

http://dx.doi.org/10.1364/OE.16.002942

Acrobat PDF (619 KB)

### Abstract

We present finite-difference time-domain (FDTD) calculations of the forces and torques on dielectric particles of various shapes, held in one or many Gaussian optical traps, as part of a study of the physical limitations involved in the construction of micro- and nanostructures using a dynamic holographic assembler (DHA). We employ a full 3-dimensional FDTD implementation, which includes a complete treatment of optical anisotropy. The Gaussian beams are sourced using a multipole expansion of a fifth order Davis beam. Force and torques are calculated for pairs of silica spheres in adjacent traps, for silica cylinders trapped by multiple beams and for oblate silica spheroids and calcite spheres in both linearly and circularly polarized beams. Comparisons are drawn between the magnitudes of the optical forces and the Van der Waals forces acting on the systems. The paper also considers the limitations of the FDTD approach when applied to optical trapping.

© 2008 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. E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. **72**, 1810–1816 (2001). [CrossRef]

3. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**, 169–175 (2002). [CrossRef]

4. Y. Roichman and D. G. Grier, “Holographic assembly of quasicrystalline photonic heterostructures,” Opt. Express **13**, 5434–5439 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-14-5434. [CrossRef] [PubMed]

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

5. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. **28**, 872–874 (2003). [CrossRef] [PubMed]

6. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express **14**, 6604–6612 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-15-6604. [CrossRef] [PubMed]

7. D. A. White, “Vector finite element modeling of optical tweezers,” Comput. Phys. Commun. **128**, 558–564 (2000). [CrossRef]

8. N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. **204**, 19–86 (1993). [CrossRef]

9. M. I. Mishchenko, L. D. Travis, and D.W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transfer **55**, 535–575 (1996). [CrossRef]

10. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. **142**, 468–471 (2001). [CrossRef]

11. S. H. Simpson and S. Hanna, “Numerical calculation of interparticle forces arising in association with holo-graphic assembly,” J. Opt. Soc. Am. A **23**, 1419–1431 (2006). [CrossRef]

12. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A **24**, 430–443 (2007). [CrossRef]

13. S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A **76**, Art. No. 043,408 (2007). [CrossRef]

14. D. W. Zhang, X. C. Yuan, S. C. Tjin, and S. Krishnan, “Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping,” Opt. Express **12**, 2220–2230 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2220. [CrossRef] [PubMed]

15. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” Opt. Express **13**, 2321–2336 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2321. [CrossRef] [PubMed]

16. R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express **13**, 3707–3718 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3707. [CrossRef] [PubMed]

## 2. Method

**E**) and magnetic (

**H**) fields propagating in a system of dielectric particles and optical traps. In FDTD simulations the

**E**and

**H**fields are updated by numerical integration of Maxwell’s curl equations in the time domain. The simulation space is divided into a lattice of discrete cells which contain the

**E**and

**H**field components. In our implementation we employed the cubic Yee cell [17

17. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. **14**, 302–307 (1966). [CrossRef]

**E**components are positioned on the edges of the cell and the

**H**components at the centers of the faces. The simulation progresses by leapfrogging the solutions for

**E**and

**H**in time and space. Different objects are represented on the grid by specifying the refractive indices of the appropriate Yee cells. Two types of non-absorbing dielectrics were used in the simulations: isotropic and negatively-birefringent, anisotropic materials. The background medium in each case was water. The isotropic material was given the refractive index of silica and the anisotropic material was modeled as quartz. The main parameters used in the simulations are summarized in Table 1. The wavelength chosen matches that of our holographic tweezer system. Unless stated otherwise, we consider the beam to propagate parallel to the

*z*-axis of the simulation box, and to be polarized in the

*x*-

*z*plane.

### 2.1. Sourcing the electric and magnetic fields

**E**and

**H**fields in the simulation space. The conventional approach to sourcing an FDTD simulation is to update the

**E**and

**H**field components across a plane in such a way that the required field is reproduced on one side of it. The most frequent choice of source is a plane wave, in which case this strategy results in the so-called ‘soft source’ condition [18].

**E**and

**H**take on the surface of any smooth plane that partitions space. This is manifested, for example, in the field equivalence theorem [19] and the orthogonality conditions of plane wave spectra [19, 20]. In each of these cases the fields on the empty side of the boundary plane are thought of as being generated by fictitious complex currents lying in the surface, which are given by:

**n̂**is the surface normal, and the incident fields,

**E**

^{inc}and

**H**

^{inc}, representing the Gaussian beam are known. In the present case, the beam propagates parallel to the z axis, and is sourced across a horizontal plane near the top of the simulation box, i.e.

**n̂**=-

**k**. The currents may then be incorporated into Maxwell’s curl equations:

*ω*is the frequency of the beam and

*θ*(

*x*,

*y*,

*z*) is the phase of the current source at a particular point which will vary across the source plane in such a way as to create the desired beam. Clearly

**M**

^{inc}is not physical but is applied in the same way as

**J**

^{inc}:

**B**with

**E**and

**D**with

**H**are generated in the usual way [18].

**E**

^{inc}and

**H**

^{inc}are required. Although the paraxial wave equation describes the propagation of Gaussian beams, in fact, solutions to the paraxial wave equation are not appropriate for use as a source in the present case. This is because the

**E**

^{inc}and

**H**

^{inc}fields derived from the paraxial wave equation are not strictly solutions of Maxwell’s equations, with the result that they do not propagate correctly in the FDTD scheme. One way of ensuring that

**E**

^{inc}and

**H**

^{inc}

*are*solutions of Maxwell’s equations is to use expansions in sets of eigenfunctions of the vector Helmholtz equation. This guarantees that, at whatever point the expansions are truncated, the resulting fields remain solutions of Maxwell’s equations. Truncation of the expansion also yields fields that can be evaluated quickly and are easy to manipulate. Here, an expansion in vector spherical wave functions (VSWFs) is used:

*a*and

_{nm}*b*are complex expansion coefficients, Rg

_{nm}**M**

_{nm}and Rg

**N**

_{nm}are regularised VSWFs [21] and

*k*is the modulus of the wavevector in the background medium. Since Rg

**M**

_{nm}and Rg

**N**

_{nm}are curls of one another the expansion for

**H**

^{inc}is immediately given:

22. 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]

*x*axis, we use:

*s*=1/

*kw*

_{0}where

*w*

_{0}corresponds to the beam waist radius. For a right- or left-circularly polarized beam, only the coefficients with

*m*=+1 or

*m*=-1, respectively, are required.

### 2.2. Calculation of forces and torques

**T͂**is the Maxwell stress tensor, defined by:

**E**and

**H**do not coincide in the Yee cell, the magnitudes of the field components at each sample point were obtained by interpolation of the eight nearest values. The forces and torques were evaluated under conditions of steady state, which occurred after between 14 and 100 optical time periods, depending on the size of the system. For the purpose of the present study, convergence was taken to be when the incremental change in force was less than 1% i.e. |Δ

*F*|/|

*F*|<0.01, when averaged over the previous eight cycles.

### 2.3. Anisotropic materials

23. J. A. Pereda, A. Vegas, and A. Prieto, “An improved compact 2D fullwave FDFD method for general guided wave structures,” Microwave Opt. Technol. Lett. **38**, 331–335 (2003). [CrossRef]

*. The electric displacement field,*

**ε****D**, is updated using the standard time stepping equations and then

**E**is calculated using:

*=*

**ε****κ**

^{-1}. All three components of

**D**are needed to calculate a single component of

**E**. However, only one of the components of

**D**is evaluated at the same location as the corresponding component of

**E**; the other components are distributed around the Yee cell. Therefore, the remaining components needed in Eq. (11) are interpolated, as shown here for

*E*(

_{x}*i*,

*j*,

*k*):

*i*,

*j*and

*k*are the grid indices of the cell. Similar relations hold for the other components.

### 2.4. The FDTD code

24. S. H. Simpson and S. Hanna, “Analysis of the effects arising from the near-field optical microscopy of homogeneous dielectric slabs,” Optics Commun. **196**, 17–31 (2001). [CrossRef]

25. S. H. Simpson and S. Hanna, “Scanning near-field optical microscopy of metallic features,” Optics Commun. **256**, 476–488 (2005). [CrossRef]

## 3. Results

### 3.1. Grid and source optimisation

14. D. W. Zhang, X. C. Yuan, S. C. Tjin, and S. Krishnan, “Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping,” Opt. Express **12**, 2220–2230 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2220. [CrossRef] [PubMed]

16. R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express **13**, 3707–3718 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3707. [CrossRef] [PubMed]

*µ*m diameter silica sphere placed along the axis of a linearly polarized Gaussian trap, for several different cell sizes, Δ. It can be seen that the forces are similar for each cell size and that, once the cell size is less than

*λ*/40, there is no substantial change in the values of the forces. Thus there appears little advantage in reducing the cell dimensions to less than

*λ*/40.

*µ*m

^{2}, i.e. 360×360 cells or 9

*λ*×9

*λ*), was felt to be a reasonable compromise between accuracy and computational expense. The largest source area considered required a threefold increase in computer time, for only a 2.5% improvement in the accuracy of the force.

### 3.2. Single trap stiffness

*µ*m diameter silica sphere from its equilibrium trapping position in a Gaussian trap.

*z*is parallel to the beam axis and the beam is polarized in the

*x-z*plane. The trap stiffness is slightly larger parallel to the

*x*-axis than parallel to the

*y*-axis. This appears to be because the irradiance of a general linearly polarized Gaussian beam is not circularly symmetric, even though the intensity will be [22

22. 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]

### 3.3. Two spheres in parallel traps

*µ*m diameter silica particles in parallel traps. Initially the equilibrium trapping position for a single particle in a single trap was found. The particle was fixed in this position relative to the trap, and an identical second particle and trap were introduced into the simulation. The two traps were mutually coherent. The excess force on each particle was calculated as a function of sphere separation (see Fig. 3).

11. S. H. Simpson and S. Hanna, “Numerical calculation of interparticle forces arising in association with holo-graphic assembly,” J. Opt. Soc. Am. A **23**, 1419–1431 (2006). [CrossRef]

*µ*m diameter spheres predict excess forces that are an order of magnitude larger than those for the 1

*µ*m sphere, and of opposite sign. This difference arises because the smaller particles are more efficient Rayleigh scatterers, which scatter a large amount of the incident light beam in a perpendicular direction to the beam axis. The implication of this finding is clear: although forces due to scattered light are generally negligible compared with trapping forces, they can become significant when the sphere separation is less than ~ 200nm and, potentially, could interfere with the assembly processes.

### 3.4. A cylinder trapped in multiple beams

26. R. Agarwal, K. Ladavac, Y. Roichman, G. H. Yu, C. M. Lieber, and D. G. Grier, “Manipulation and assembly of nanowires with holographic optical traps,” Opt. Express **13**, 8906–8912 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-8906. [CrossRef] [PubMed]

*total beam power*of 10mW; thus it is the distribution of light intensity along the cylinder that is being compared.

*z*and

*y*directions results from the fact that the intensity gradient in the beam direction is much less than that in the cross section of the beam. Comparing the spring constants in the

*x*-direction, it can be seen that the system has a higher stiffness when the laser power is concentrated towards the ends of the cylinder, i.e. when only two beams are used, decreasing as the number of beams increases.

*z*direction, because only light incident close to the ends of the cylinder contributes to the gradient force in the

*x*direction. As a result, the rod is able to slide relatively easily along its long axis. For example, Fig. 5 shows the restoring force as the cylinder is displaced along the

*x*axis in a system with three traps. At zero displacement, there is no restoring force, corresponding to the position shown in Fig. 4. The restoring force is a maximum at a displacement of ~ 400 nm, as one trap is now at the end of the cylinder. Clearly, to reduce the extent of this sliding the outermost traps should be positioned as close as possible to the ends of the cylinder.

*y*axis i.e. the axis perpendicular to the beam and rod major axis, and the resulting torque on the particle is shown. For angles less than 45°, the beams apply a restoring torque in all cases, indicating that the horizontal configuration is stable. In the case of the two beam simulation the torque remains negative at larger angles, as there is little light intensity between the beams. This differs from the three beam case where the central beam causes the torque to change sign when the cylinder is at about 45°, stabilizing the vertical orientation when the rod is rotated by more than 45°. The simulations with four and five beams show a smaller magnitude of torque because the light is distributed along the cylinder and the intensity gradients are much reduced.

### 3.5. Oblate silica particles trapped in a circularly polarized beam

12. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A **24**, 430–443 (2007). [CrossRef]

*δ*=0.2.

*h̄ω*per photon, it is presumed that the interaction is mediated by an orbital component as was suggested in [12

12. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A **24**, 430–443 (2007). [CrossRef]

### 3.6. Trapping of anisotropic spheres

*µ*m diameter calcite sphere with the angle,

*ϕ*, between the optic axis and polarization direction in a linearly polarized Gaussian trap. The angle,

*ϕ*, is measured in a plane perpendicular to the beam axis, and the torque is calculated about the beam axis. The results show a sin2

*ϕ*dependence, which compares favourably with previous work on geometrically anisotropic particles that made used of the T-matrix formalism [13

13. S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A **76**, Art. No. 043,408 (2007). [CrossRef]

27. M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E72, Art. No. 056,610 (2005). [CrossRef]

28. V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transfer **106**, 274–284 (2007). [CrossRef]

*w*

_{0}, i.e. up to

*ca*. 450nm in the figure, the torque scales approximately with the volume of the particle. However, for particles with radius greater than

*w*

_{0}the lateral edges of the particle, which are outside the beam, contribute little to the torque and so the torque varies linearly with the radius of the particle.

## 4. Discussion

*λ*/40 was selected as optimal, on the grounds that the computed optical forces on 1

*µ*m diameter silica spheres appear to converge at this lattice dimension. The surface roughness arising from the use of the lattice in this case is around 1/60 of the sphere diameter and may be regarded as negligible. However, in the simulations of silica cylinders presented in Section 3.4, the use of a 15 nm cell dimension is quite crude, and the cylindrical cross section is far from circular. For the purposes of the present study, this inaccuracy was deemed not to be important.

*µ*m diameter particle, an excess force of 0.02pN corresponds to a horizontal displacement of 1 nm or a vertical displacement of 7 nm from the equilibrium trapping position. However, at very small separations (≲ 200nm), the excess force increases dramatically, and this effect is compounded when smaller particles are considered. At close separations, the excess optical force,

*F*, can be comparable with the Van der Waals attraction, and may interfere with attempts to assemble structures. For example, taking the Hamaker constant for silica spheres in water to be

_{x}*A*

_{131}=0.8×10

^{-20}J, gives an attractive force of 0.033pN for 0.5

*µ*m diameter spheres, or 0.067pN for 1

*µ*m diameter spheres separated by 100 nm, compared with an optically induced 0.08pN repulsion for the 1

*µ*m spheres and 0.17pN attraction for the 0.5

*µ*m spheres. Although we have assumed the trap geometry to be vertical, no account has been taken of the gravitational force on the silica spheres. In fact the force on a 1

*µ*m diameter sphere in water will be 0.007 pN. This will lead to a vertical change in trapping position of ~ 2 nm, and may safely be ignored.

*ϕ*dependance of torque on setting angle (Fig. 9(a)) provides a satisfying demonstration that optical anisotropy leads to the same behaviour as geometrical anisotropy when particles are held in linearly polarized traps, though not when circularly polarized traps are used. It also serves to validate the operation of our anisotropic FDTD code. That the orientation of anisotropic particles can be controlled by the beam polarisation has been well known for many years, and provides another means of orientational control that may be exploited when assembling structures in the DHA. The occurrence of torque reversal for geometrically anisotropic particles in circularly polarized beams has now been predicted using both T matrix and FDTD formalisms. The effect is quite subtle, depending on the overlap of the particle with the perimeter of the trap, but is large compared with any numerical errors occurring during integration of the Maxwell stress tensor. We are currently fabricating oblate spheroids with which to test the prediction experimentally. The absence of torque reversal for optically anisotropic particles is intriguing and is the subject of further studies.

## 5. Conclusions

*µ*m diameter spheres, we have shown that a FDTD grid cell size of 15nm or

*λ*/40 is adequate for the calculation of accurate forces and torques. Finer grids will be needed when dealing with smaller systems. A method has been shown for sourcing the simulation with a Gaussian beam, by means of a multipole expansion of a fifth order Davis beam. The dimensions of the source plane are important in determining the absolute values of the forces calculated. By way of a compromise between computational expense and accuracy, we have standardised on a source plane measuring 9

*λ*×9

*λ*or 360×360 cells.

*δ*=0.2). Optically anisotropic particles behave in the same manner as geometrically anisotropic particles in linearly polarized beams, displaying the same sin 2

*ϕ*torque dependence. However, optically anisotropic particles do not appear to display torque reversal in circularly polarized beams at any size scale so far simulated.

## Acknowledgements

## References and links

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

2. | E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. |

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

4. | Y. Roichman and D. G. Grier, “Holographic assembly of quasicrystalline photonic heterostructures,” Opt. Express |

5. | J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. |

6. | C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express |

7. | D. A. White, “Vector finite element modeling of optical tweezers,” Comput. Phys. Commun. |

8. | N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. |

9. | M. I. Mishchenko, L. D. Travis, and D.W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transfer |

10. | T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. |

11. | S. H. Simpson and S. Hanna, “Numerical calculation of interparticle forces arising in association with holo-graphic assembly,” J. Opt. Soc. Am. A |

12. | S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A |

13. | S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A |

14. | D. W. Zhang, X. C. Yuan, S. C. Tjin, and S. Krishnan, “Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping,” Opt. Express |

15. | A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” Opt. Express |

16. | R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express |

17. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. |

18. | A. Taflove and S. C. Hagness, |

19. | S. A. Schelkunoff, |

20. | P. Clemmow, |

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

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

23. | J. A. Pereda, A. Vegas, and A. Prieto, “An improved compact 2D fullwave FDFD method for general guided wave structures,” Microwave Opt. Technol. Lett. |

24. | S. H. Simpson and S. Hanna, “Analysis of the effects arising from the near-field optical microscopy of homogeneous dielectric slabs,” Optics Commun. |

25. | S. H. Simpson and S. Hanna, “Scanning near-field optical microscopy of metallic features,” Optics Commun. |

26. | R. Agarwal, K. Ladavac, Y. Roichman, G. H. Yu, C. M. Lieber, and D. G. Grier, “Manipulation and assembly of nanowires with holographic optical traps,” Opt. Express |

27. | M. K. Liu, N. Ji, Z. F. Lin, and S. T. Chui, “Radiation torque on a birefringent sphere caused by an electromagnetic wave,” Phys. Rev. E72, Art. No. 056,610 (2005). [CrossRef] |

28. | V. L. Y. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transfer |

**OCIS Codes**

(090.5694) Holography : Real-time holography

(290.5825) Scattering : Scattering theory

**ToC Category:**

Trapping

**History**

Original Manuscript: January 4, 2008

Revised Manuscript: February 8, 2008

Manuscript Accepted: February 13, 2008

Published: February 19, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

David C. Benito, Stephen H. Simpson, and Simon Hanna, "FDTD simulations of forces on particles
during holographic assembly," Opt. Express **16**, 2942-2957 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-2942

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### References

- A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970). [CrossRef]
- E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001). [CrossRef]
- J. E. Curtis, B. A. Koss and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002). [CrossRef]
- Y. Roichman and D. G. Grier, "Holographic assembly of quasicrystalline photonic heterostructures," Opt. Express 13, 5434-5439 (2005). [CrossRef] [PubMed]
- J. E. Curtis and D. G. Grier, "Modulated optical vortices," Opt. Lett. 28, 872-874 (2003). [CrossRef] [PubMed]
- C. H. J. Schmitz, K. Uhrig, J. P. Spatz and J. E. Curtis, "Tuning the orbital angular momentum in optical vortex beams," Opt. Express 14, 6604-6612 (2006). [CrossRef] [PubMed]
- D. A. White, "Vector finite element modeling of optical tweezers," Comput. Phys. Commun. 128, 558-564 (2000). [CrossRef]
- N. V. Voshchinnikov and V. G. Farafonov, "Optical properties of spheroidal particles," Astrophys. Space Sci. 204, 19-86 (1993). [CrossRef]
- M. I. Mishchenko, L. D. Travis and D.W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: A review," J. Quant. Spectrosc. Radiat. Transfer 55, 535-575 (1996). [CrossRef]
- T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001). [CrossRef]
- S. H. Simpson and S. Hanna, "Numerical calculation of interparticle forces arising in association with holographic assembly," J. Opt. Soc. Am. A 23, 1419-1431 (2006). [CrossRef]
- S. H. Simpson and S. Hanna, "Optical trapping of spheroidal particles in Gaussian beams," J. Opt. Soc. Am. A 24, 430-443 (2007). [CrossRef]
- S. H. Simpson, D. C. Benito and S. Hanna, "Polarization-induced torque in optical traps," Phys. Rev. A 76, 408 (2007). [CrossRef]
- D. W. Zhang, X. C. Yuan, S. C. Tjin and S. Krishnan, "Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping," Opt. Express 12, 2220-2230 (2004). [CrossRef] [PubMed]
- A. R. Zakharian, M. Mansuripur and J. V. Moloney, "Radiation pressure and the distribution of electromagnetic force in dielectric media," Opt. Express 13, 2321-2336 (2005). [CrossRef] [PubMed]
- R. C. Gauthier, "Computation of the optical trapping force using an FDTD based technique," Opt. Express 13, 3707-3718 (2005). [CrossRef] [PubMed]
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