## Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles |

Optics Express, Vol. 19, Issue 17, pp. 16526-16541 (2011)

http://dx.doi.org/10.1364/OE.19.016526

Acrobat PDF (1634 KB)

### Abstract

The accuracy of the discrete dipole approximation (DDA) for computing forces and torques in optical trapping experiments is discussed in the context of dielectric spheres and a range of low symmetry particles, including particles with geometric anisotropy (spheroids), optical anisotropy (birefringent spheres) and structural inhomogeneity (core-shell spheres). DDA calculations are compared with the results of exact T-matrix theory. In each case excellent agreement is found between the two methods for predictions of optical forces, torques, trap stiffnesses and trapping positions. Since the DDA lends itself to calculations on particles of arbitrary shape, the study is augmented by considering more general systems which have received recent experimental interest. In particular, optical forces and torques on low symmetry letter-shaped colloidal particles, birefringent quartz cylinders and biphasic Janus particles are computed and the trapping behaviour of the particles is discussed. Very good agreement is found with the available experimental data. The efficiency of the DDA algorithm and methods of accelerating the calculations are also discussed.

© 2011 OSA

## 1. Introduction

1. E. Fällman and O. Axner, “Influence of a glass-water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. **42**, 3915–3926 (2003). [CrossRef] [PubMed]

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

3. D. A. White, “Numerical modeling of optical gradient traps using the vector finite element method,” J. Comp. Phys. **159**, 13–37 (2000). [CrossRef]

4. R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Optics Express13, 3707–3718 (2005). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3707. [CrossRef] [PubMed]

5. D. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express16, 2942–2957 (2008). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-2942. [CrossRef] [PubMed]

6. P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E **72**, 046708 (2005). [CrossRef]

11. V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. **112**, 1711–1725 (2011). [CrossRef]

12. 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]

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

15. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. **79**, 775–824 (2003). [CrossRef]

15. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. **79**, 775–824 (2003). [CrossRef]

16. V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. **110**, 1460–1471 (2009). [CrossRef]

9. S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A **27**, 1255–1264 (2010). [CrossRef]

10. L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. **108**(7), 073110 (2010). [CrossRef]

17. J. N. Wilking and T. G. Mason, “Multiple trapped states and angular Kramers hopping of complex dielectric shapes in a simple optical trap,” Europhys. Lett. **81**, 58005 (2008). [CrossRef]

18. C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods **4**, 223–225 (2007). [CrossRef] [PubMed]

19. B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. **474**, 377–404 (2010). [CrossRef]

20. I. Kretzschmar and J. H. K. Song, “Surface-anisotropic spherical colloids in geometric and field confinement,” Curr. Opin. Colloid Interface Sci. **16**, 84–95 (2011). [CrossRef]

## 2. Methods

### 2.1. T-matrix theory

**M**

*and Rg*

_{n,m}**N**

*are regularised VSWFs. Application of appropriate boundary conditions leads to a linear relationship between the expansion coefficients for the incident and scattered fields, i.e. the T-matrix, thereby solving the scattering problem. For homogeneous and isotropic spheres, the T-matrix is diagonal, with entries given by the Mie coefficients. For non-spherical objects we calculate the T-matrix using the exact, extended boundary condition method (EBCM). Optical anisotropy is incorporated by using the VSWF basis set to construct a new basis in the anisotropic material such that the divergence of the electric displacement vector*

_{n,m}**D**vanishes, while simultaneously satisfying an appropriate wave equation [21

21. Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E **69**, 056614 (2004). [CrossRef]

22. S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A **26**, 625–638 (2009). [CrossRef]

24. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A **16**, 1381–1386 (1999). [CrossRef]

9. S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A **27**, 1255–1264 (2010). [CrossRef]

*m*is restricted to ±1 and

*I*is given by the following, which is evaluated numerically:

_{n,m}*z*

_{0}is the position of the complex source and also determines the beam waist [24

24. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A **16**, 1381–1386 (1999). [CrossRef]

*κ*is the wavenumber;

*κz*

_{0}= 3.0 has been used throughout, corresponding to a beam with waist radius

*λ*/2.

*γ*is given by:

_{n,m}### 2.2. Discrete Dipole Approximation

25. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**(4), 1491–1499 (1994). [CrossRef]

26. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. **106**, 558–589 (2007). [CrossRef]

9. S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A **27**, 1255–1264 (2010). [CrossRef]

*δ*. The field acting on any particular cell is given by the sum of the incident field and that scattered from each of the other cells. The field scattered by each cell is dipolar, with a strength determined by the product of the total field acting on the cell and its polarizability. These relationships may be written as a large, dense set of linear equations that are solved iteratively to give the polarization of each cell.

*in vacuo*. Optical trapping experiments, on the other hand, are usually performed in a medium such as water. By the scale invariance rule, the optical scattering for a fixed geometry depends on the wavenumbers in the different materials rather than the absolute values of the refractive indices [13]. Therefore the polarizability of the individual cells,

*α*, may be written as [26

26. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. **106**, 558–589 (2007). [CrossRef]

*α*has been corrected to ensure compatibility with the optical scattering theorem, and

*α*

_{CM}is the Claussius-Mossotti polarizability of the scatterer (permittivity

*ɛ*

_{s}), relative to the ambient medium (permittivity

*ɛ*

_{m}).

*a*is the radius of a sphere whose volume is equal to one cell of the DDA lattice. This formulation may be extended to deal with optical anisotropy [27

27. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. **333**, 848–872 (1988). [CrossRef]

6. P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E **72**, 046708 (2005). [CrossRef]

7. P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. **101**, 023106 (2007). [CrossRef]

**p**is the polarization of the cell obtained by solving the DDA equations and

**E**is the total electric field at the cell. In Eq. (6), subscripts denote Cartesian components and repeated indices are summed. The angle brackets indicate cycle averages. The cell torque 〈

**T**

^{cell}〉 consists of two contributions, an orbital part associated with the force on the cell,

**r**× 〈

**F**

^{cell}〉, and a spin part,

*p*obtained from the DDA calculation, using the

_{i}*α*of Eq. (5), must be multiplied by

*ɛ*

_{m}[23]. An alternative formulation of radiation forces on individual dipoles in the DDA is given by Hoekstra et al. [28

28. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete dipole approximation,” J. Opt. Soc. Am. A **18**, 1944–1953 (2001). [CrossRef]

*per se*, but convergence on continuum results has been rigorously justified [26

26. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. **106**, 558–589 (2007). [CrossRef]

29. B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains.1. Superthermal spin-up,” Astrophys. J. **470**, 551–565 (1996). [CrossRef]

7. P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. **101**, 023106 (2007). [CrossRef]

### 2.3. Stiffness coefficients

**F**is the generalized force, (

*F*,

_{x}*F*,

_{y}*F*,

_{z}*τ*,

_{x}*τ*,

_{y}*τ*), and

_{z}**q**the generalized coordinates (

*x,y,z*,

*θ*,

_{x}*θ*,

_{y}*θ*), where the vector

_{z}**= (**

*θ**θ*,

_{x}*θ*,

_{y}*θ*) indicates a rotation of |

_{z}**| radians about an axis**

*θ***.**

θ ^ **q**

_{eqm}is the equilibrium trapping point. The stiffness matrix,

**K**, decomposes into a 2 × 2 array of tensors: in which the tensors

**K**and

^{tt}**K**relate forces to displacements and torques to rotations, while the pseudo-tensors

^{rr}**K**and

^{tr}**K**couple rotations to forces, and displacements to torques, e.g.:

^{rt}**K**need not be symmetric. This has implications for the thermal motion of the trapped particle, and for force and torque calibration. In general,

**K**describes the behavior of forces close to equilibrium and, together with the hydrodynamic resistance, determines the stability of the equilibrium configuration. It plays an essential role whenever optical traps are used quantitatively. Because of its importance, and because it captures the local characteristics of the force field, we use it as a basis for comparing T-matrix and DDA calculations.

### 2.4. Qualitative description of optical forces

31. 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]

22. S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A **26**, 625–638 (2009). [CrossRef]

**f**〉 consists of two terms. The first relies on the electric current,

**J**, and therefore depends on optical absorption to operate. The second includes the gradient of the permittivity, ∇

*ɛ*, and incorporates the effects of inhomogeneity and material boundaries. For an optically isotropic, non-absorbing dielectric, the first term vanishes and the force density reduces to: i.e. the force density is concentrated on the surface and is normal to it.

_{jk}**t**〉 also contains two terms. The first,

**r**× 〈

**f**〉, depends on the distribution of 〈

**f**〉 over the particle. For a non-absorbing, optically isotropic material the fact that the force is normal to the surface implies that the effective body torque must be zero parallel to an axis of cylindrical symmetry i.e. an optical field cannot apply a torque to a sphere of this sort, while an optically isotropic cylinder diffuses freely about its axis. The second term, 〈

**D**×

**E**〉, depends on optical anisotropy. For optically isotropic media, it will be zero. Otherwise there will be a tendency to align the greatest component of the refractive index tensor with

**E**.

**f**〉 and 〈

**t**〉, should not be confused with the contributions arising from individual cells in the DDA simulation, 〈

**F**

^{cell}〉 and 〈

**T**

^{cell}〉 [Eqs. (6) and (7)]. Although both sets of expressions may be integrated to yield the total body force and torque, they clearly represent different quantities and have different properties. For example, by Eq. (14), 〈

**f**〉 is confined to the surface of an optically isotropic, non-absorbing dielectric and is normally directed. On the other hand, 〈

**F**

^{cell}〉 need not be zero for a DDA cell in the interior of a modelled object nor, for a boundary cell, need it be normal to the surface on which it lies. It is one of the aims of the present study to demonstrate that the sums of 〈

**F**

^{cell}〉 and 〈

**T**

^{cell}〉 over the DDA cells do indeed converge on the theoretical values of force and torque as given rigorously by the T-matrix theory, and that this convergence can be reached without incurring excessive computational cost, for arbitrary particles of colloidal size and medium relative refractive index.

## 3. Numerical Results

*z*-axis coincident with the beam axis, and the

*x*-axis parallel to the polarization direction. Propagation is in the positive

*z*-direction and the ambient medium is water, with refractive index

*n*

_{m}= 1.333, in which the laser wavelength is 0.6

*μ*m.

### 3.1. Dielectric spheres

**K**is non-zero. This tensor is diagonal in the indicated coordinate frame (Fig. 1), and contains three independent stiffness coefficients,

^{tt}*k*,

_{x}*k*and

_{y}*k*, relating to restoring forces parallel to each of the axes. Figure 2 shows the results of DDA calculations of the trapping height,

_{z}*z*

_{eqm}, and the stiffness coefficient,

*k*, for dielectric spheres of radii,

_{x}*r*= 100, 200,300 and 400nm and refractive index

*n*

_{s}= 1.1

*n*

_{m}, as a function of the DDA lattice parameter,

*δ*. The results for large

*δ*are erratic, but ultimately settle on particular values of

*z*

_{eqm}and

*k*when

_{x}*δ*≲ 25 nm. The most substantial error occurs in

*z*

_{eqm}for the smaller spheres. Based on these results, we use

*δ*= 15nm in the work presented below, unless stated otherwise.

*N*, used in Fig. 2 scales as expected with

*r*

^{3}and with

*δ*

^{−3}.

*N*ranges from 19, for

*r*= 100 nm and

*δ*= 60 nm to 267,731 for

*r*= 400 nm and

*δ*= 10 nm. The calculation times increase with

*N*, the slowest being 64 s per force evaluation, when running in parallel on an 8 core, 2.5 GHz Xeon system. These times should be compared with a few seconds for a T-matrix calculation, or several hours for an equivalent FDTD simulation [5

5. D. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express16, 2942–2957 (2008). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-2942. [CrossRef] [PubMed]

*z*

_{eqm}, and translational stiffness parallel to the polarisation direction,

*k*. In Figs. 3a and 3b, the effect of lattice spacing,

_{x}*δ*, is assessed. It can be seen that the overall shape of the curves is preserved for quite large values of

*δ*, but that they become increasingly noisy for

*δ*> 20nm. Figures 3c and 3d show the same curves for DDA calcuations with

*δ*= 15nm compared with exact T-matrix calculations of the same system. The DDA and T-matrix results are so similar that they are visually indistinguishable. The greatest difference observed across the range of radii is

*c.*1%, while the average difference is

*c.*0.2%. Ambient media with three different values of

*n*

_{m}have been used confirming the need to incorporate

*ɛ*

_{m}in the DDA force calculation as described above (Section 2.2).

33. A. B. Stilgoe, T. A. Nieminen, G. Knoner, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express16, 15039–15051 (2008). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2661. [CrossRef] [PubMed]

### 3.2. Geometric Anisotropy

**27**, 1255–1264 (2010). [CrossRef]

22. S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A **26**, 625–638 (2009). [CrossRef]

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

*x*or

*y*axes, but not the

*z*axis, will give rise to restoring torques.

*x*or

*y*directions also induce excess forces parallel to the beam axis, as well as coupling torques. The forces induced are even functions of displacement, while the coupling torques are odd functions. Similarly, rotations about the

*x*and

*y*axes produce excess axial forces (even in displacement) and coupling forces (odd in displacement). Forces that are even functions of displacement do not appear in the stiffness matrix. Therefore, many of the components of

**K**vanish identically. The complete set of forces and torques acting on a prolate spheroid are analogous to those acting on a cylinder [30].

**K**and

^{tt}**K**will be diagonal, while

^{rr}**K**and

^{rt}**K**include coefficients that relate displacements in

^{tr}*x*and

*y*directions to torques about the

*y*and

*x*axes, and rotations about the

*x*and

*y*axes to forces in the

*y*and

*x*directions, respectively.

**K**, for prolate spheroids (

*n*

_{s}= 1.45) in water. The spheroids have an aspect ratio

*b/a*= 2 and the results are plotted as a function of equivalent radius

*δ*= 10nm. Once again the DDA results are practically indistinguishable from the T-matrix results. The translational stiffness coefficients (Fig. 4a) behave similarly to those of a sphere: when the particle is too small to scatter strongly or big enough to envelope the focal region completely, the translational stiffness is low. Between these regimes, a maximum is reached. For the range of sizes considered, the rotational stiffness coefficients (Fig. 4b) increase monotonically with the size of the particle. This is due, in part, to the fact that rotation through a small angle,

*δθ*, involves increasingly large displacements of the tip of the object, as the spheroid increases in height. This effect counterbalances the reduction in rotational stiffness one might expect to occur when the particle encompasses the focal region.

**K**is non-symmetric [30]. This is confirmed by Figs. 4c and 4d, from which it is clear that

17. J. N. Wilking and T. G. Mason, “Multiple trapped states and angular Kramers hopping of complex dielectric shapes in a simple optical trap,” Europhys. Lett. **81**, 58005 (2008). [CrossRef]

*xz*-plane). Therefore, attention is restricted to translation and rotation in this plane. The forces and torques experienced by the letters, in the vicinity of the experimentally established trapping configurations, are examined.

*y*through two pivot points, and the induced torque is plotted as a function of rotation angle,

*θ*. The first pivot is located at the centre of the letter, on the diagonal, while the second is half way up the left hand vertical strut. As can be seen, rotation about the central pivot has a stable orientation at

_{y}*θ*≈ 0.2

_{y}*π*rads. This corresponds to the diagonal strut of the ‘N’ being aligned with the beam axis. Rotation about the left hand pivot produces a stable orientation just off vertical (

*θ*≈ −0.05

_{y}*π*rads). The two trapping orientations are illustrated schematically in Fig. 5d.

17. J. N. Wilking and T. G. Mason, “Multiple trapped states and angular Kramers hopping of complex dielectric shapes in a simple optical trap,” Europhys. Lett. **81**, 58005 (2008). [CrossRef]

*θ*≈ ±

_{y}*π*/2 rads (see Fig. 5d), as found experimentally.

*x-*component of the optical force as the letter ‘O’ is displaced in the

*x-*direction. The solid curve is generated when the letter is oriented vertically, the dashed curve when its long axis points in the

*x-*direction. As was observed experimentally [17

**81**, 58005 (2008). [CrossRef]

### 3.3. Optical Anisotropy

*ɛ*will be zero inside a homogeneous birefringent sphere, and 〈

**f**〉 will be confined to the surface and normal to it. As a result the boundary contribution to the torque vanishes identically. Therefore, 〈

**t**〉 is entirely given by the second, spin-like, term in Eq. (13): 〈

**D**×

**E**〉. Since positive birefringence leads to the alignment of the optic axis with the polarization direction, this property allows control over the rotation of optically trapped birefringent particles [35

35. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature **394**, 348–350 (1998). [CrossRef]

*θ*) [36

_{z}36. S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A **76**, 043408 (2007). [CrossRef]

**t**〉 depends on the phase difference between

**E**and

**D**. For large path lengths the sign of 〈

**t**〉 changes and so the magnitude of the torque, obtained by integrating 〈

**t**〉 over the particle [22

**26**, 625–638 (2009). [CrossRef]

*ɛ*= 0.4.

*x*. Such particles have applications in novel torque microscopes [18

18. C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods **4**, 223–225 (2007). [CrossRef] [PubMed]

19. B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. **474**, 377–404 (2010). [CrossRef]

*n*= 0.009. The aspect ratios of the cylinders vary between

*c.*1.4 and 50. While increasing the length of the rod adds stability and increases the stiffness with which the cylinder is held vertically, it does not greatly improve the angular behavior about its symmetry axis. Indeed, it can be seen that there is a distinct change in behaviour between the narrower cylinders (0.1 ≤

*r*≤ 0.2

*μ*m) and the wider ones (0.25 ≤ r ≤ 0.35

*μ*m). For the narrower cylinders

*μ*m, and then decreases with further increases in cylinder length.

**E**and

**D**, so that 〈

**t**〉 goes negative and starts reducing the total torque i.e. the reduction in torque due to aggregated phase differences is only apparent when propagation down the rod is allowed. Consequently, decreasing

*n*

_{o}and

*n*

_{e}, while maintaining the same Δ

*n*, will lead to an increase in the cut-off radius and the reduction in torque will occur for wider rods, as has been observed (unpublished results).

*μ*m. Additional torque may be gained by increasing the radius even further, but this will become ineffective once the cylinder radius exceeds the beam waist radius. Interestingly, the quartz cylinders used in experiments have radii in the range 0.13 to 0.3

*μ*m, and lengths around 1

*μ*m, suggesting that there is scope for improving the torque that can be delivered experimentally by this method [18

18. C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods **4**, 223–225 (2007). [CrossRef] [PubMed]

19. B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. **474**, 377–404 (2010). [CrossRef]

*μ*m.rad.

^{−1}mW

^{−1}for a cylinder with radius 0.265

*μ*m and length 1.1

*μ*m [18

**4**, 223–225 (2007). [CrossRef] [PubMed]

### 3.4. Inhomogeneity

*n*= 1.35) and a high index shell (silica,

*n*= 1.45). In Fig. 7, we plot the equilibrium trapping height,

*z*

_{eqm}, and the three components of translational stiffness,

*k*,

_{x}*k*and

_{y}*k*determined at

_{z}*z*

_{eqm}, comparing the results from DDA calculations and T-matrix theory. Good agreement is observed between the two methods for

*z*

_{eqm}(see Fig. 7b), but some discrepancies are found for the stiffness results (Fig. 7c), which are particularly obvious when the shell gets thinner (see Fig. 7d). For thin shells, the discrete nature of the DDA lattice appears as discontinuities in the gradients of the stiffness curves.

*z*

_{eqm}shows an obvious variation with core size. When the core is very small, the particle behaves like a silica sphere; when it is very big, the particle is effectively a PTFE sphere. For a range of intermediate core sizes, the shell traps preferentially and, as it is confined to the edge of the particle, so the particle is displaced in the trap. In particular, for core radii between 0.35 and 0.55

*μ*m, trapping through the silica shell is favored to the extent that there are two distinct trapping heights (see Fig. 7b). In the first of these, the centre of the sphere is downstream of the focus (

*z*

_{eqm}> 0) and the trap is concentrated on the part of the shell nearest to the focus. In the second, the centre of the sphere is upstream the focus (

*z*

_{eqm}< 0), and the opposite region of the shell is trapped. Evaluation of the trap stiffnesses indicates that, when

*z*

_{eqm}> 0, the trap is stable for all displacements whereas, for

*z*

_{eqm}< 0, it is unstable to displacements in

*x*or

*y*. Therefore, the trap stiffnesses shown in Fig. 7c are only given for

*z*

_{eqm}> 0.

*μ*m would produce a quasi-isotropic trap with

*k*≃

_{x}*k*≃

_{y}*k*.

_{z}*xz*-section through the force field, and a circular region in the

*xy*-section (shown in black in Fig. 8) within which the force tends to zero. While the only stable equilibrium configuration is that described above, it is clear that there is a tendency for the core-shell particle to move away from the beam axis and away from the focus, in all directions, because of the need to overlap the shell with the more intense parts of the beam.

20. I. Kretzschmar and J. H. K. Song, “Surface-anisotropic spherical colloids in geometric and field confinement,” Curr. Opin. Colloid Interface Sci. **16**, 84–95 (2011). [CrossRef]

*y*-axis,

*τ*, as a function of rotation angle about the same axis,

_{y}*θ*, for a range of Janus spheres with different radii.

_{y}*θ*= 0 corresponds to the interface normal being aligned with the

_{y}*z*-axis. As can be seen, the torques experienced are complicated in structure. As the sphere radius increases the numbers of maxima and minima also increase and the stability of the

*θ*= 0 configuration switches. In fact, for most radii, the sphere is marginally stable to small rotations around

_{y}*θ*= 0. However, for larger rotations, the configuration becomes unstable for all radii. For radii of 0.6 and 0.7

_{y}*μ*m, stable rotations become apparent around

*θ*=

_{y}*π*/2 radians. These are accompanied by transverse forces,

*F*(Fig. 9c) that act to move the higher refractive index part of the Janus sphere into the beam. The behaviour for rotation about the

_{x}*x*-axis is slightly different (Fig. 9b). The configuration at

*θ*= 0 is unstable to

_{x}*x*-rotations for all sphere radii. However, the accompanying lateral force,

*F*(Fig. 9d) is two orders of magnitude smaller than the previous case. These issues will be discussed at greater length in a forthcoming article.

_{y}## 4. Discussion & Conclusions

15. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. **79**, 775–824 (2003). [CrossRef]

*c.*10

^{4}dipoles took several hours to solve for the dipole moment on an 8-core Linux workstation [11

11. V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. **112**, 1711–1725 (2011). [CrossRef]

*δ*= 20nm. The typical time required for each force calculation was ≲ 120 s, on an 8 core, 2.5 GHz Xeon system. The calculation ran comfortably within the 16GB of memory available on the system. This performance compares very favourably with reports in the literature [11

11. V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. **112**, 1711–1725 (2011). [CrossRef]

*z*

_{eqm}for each combination of parameters considered above, and a further five evaluations are then needed to determine each stiffness coefficient. Therefore, it is vital to ensure that the DDA calculations are performed efficiently. We perform the DDA matrix inversion iteratively using a Krylov method [9

**27**, 1255–1264 (2010). [CrossRef]

## Acknowledgments

## References and links

1. | E. Fällman and O. Axner, “Influence of a glass-water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. |

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

3. | D. A. White, “Numerical modeling of optical gradient traps using the vector finite element method,” J. Comp. Phys. |

4. | R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Optics Express13, 3707–3718 (2005). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3707. [CrossRef] [PubMed] |

5. | D. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express16, 2942–2957 (2008). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-2942. [CrossRef] [PubMed] |

6. | P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E |

7. | P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. |

8. | D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A: Pure Appl. Opt. |

9. | S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A |

10. | L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. |

11. | V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. |

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

13. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

14. | S. H. Simpson and S. Hanna, “Numerical calculation of inter-particle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A |

15. | F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. |

16. | V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. |

17. | J. N. Wilking and T. G. Mason, “Multiple trapped states and angular Kramers hopping of complex dielectric shapes in a simple optical trap,” Europhys. Lett. |

18. | C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods |

19. | B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. |

20. | I. Kretzschmar and J. H. K. Song, “Surface-anisotropic spherical colloids in geometric and field confinement,” Curr. Opin. Colloid Interface Sci. |

21. | Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E |

22. | S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A |

23. | J. A. Stratton, |

24. | C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A |

25. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

26. | M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. |

27. | B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. |

28. | A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete dipole approximation,” J. Opt. Soc. Am. A |

29. | B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains.1. Superthermal spin-up,” Astrophys. J. |

30. | S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E |

31. | 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. |

32. | I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep., Phys. Lett. |

33. | A. B. Stilgoe, T. A. Nieminen, G. Knoner, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express16, 15039–15051 (2008). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2661. [CrossRef] [PubMed] |

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

35. | M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature |

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

37. | M. Doi and S. F. Edwards, |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

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

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(290.5825) Scattering : Scattering theory

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: June 13, 2011

Revised Manuscript: July 13, 2011

Manuscript Accepted: July 13, 2011

Published: August 12, 2011

**Virtual Issues**

Vol. 6, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Stephen H. Simpson and Simon Hanna, "Application of the discrete dipole approximation to optical trapping calculations of inhomogeneous and anisotropic particles," Opt. Express **19**, 16526-16541 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16526

Sort: Year | Journal | Reset

### References

- E. Fällman and O. Axner, “Influence of a glass-water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. 42, 3915–3926 (2003). [CrossRef] [PubMed]
- Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996). [CrossRef]
- D. A. White, “Numerical modeling of optical gradient traps using the vector finite element method,” J. Comp. Phys. 159, 13–37 (2000). [CrossRef]
- R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Optics Express 13, 3707–3718 (2005). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3707 . [CrossRef] [PubMed]
- D. Benito, S. H. Simpson, and S. Hanna, “FDTD simulations of forces on particles during holographic assembly,” Opt. Express 16, 2942–2957 (2008). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-2942 . [CrossRef] [PubMed]
- P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005). [CrossRef]
- P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. 101, 023106 (2007). [CrossRef]
- D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A: Pure Appl. Opt. 9(8), S228–S234 (2007). [CrossRef]
- S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A 27, 1255–1264 (2010). [CrossRef]
- L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108(7), 073110 (2010). [CrossRef]
- V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. 112, 1711–1725 (2011). [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]
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).
- S. H. Simpson and S. Hanna, “Numerical calculation of inter-particle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A 23, 1419–1431 (2006). [CrossRef]
- F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. 79, 775–824 (2003). [CrossRef]
- V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009). [CrossRef]
- J. N. Wilking and T. G. Mason, “Multiple trapped states and angular Kramers hopping of complex dielectric shapes in a simple optical trap,” Europhys. Lett. 81, 58005 (2008). [CrossRef]
- C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007). [CrossRef] [PubMed]
- B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. 474, 377–404 (2010). [CrossRef]
- I. Kretzschmar and J. H. K. Song, “Surface-anisotropic spherical colloids in geometric and field confinement,” Curr. Opin. Colloid Interface Sci. 16, 84–95 (2011). [CrossRef]
- Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69, 056614 (2004). [CrossRef]
- S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009). [CrossRef]
- J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
- C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999). [CrossRef]
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- B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]
- A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete dipole approximation,” J. Opt. Soc. Am. A 18, 1944–1953 (2001). [CrossRef]
- B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains.1. Superthermal spin-up,” Astrophys. J. 470, 551–565 (1996). [CrossRef]
- S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031,141 (2010).
- 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]
- I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep., Phys. Lett. 52, 133–201 (1979).
- A. B. Stilgoe, T. A. Nieminen, G. Knoner, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express 16, 15039–15051 (2008). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2661 . [CrossRef] [PubMed]
- S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430–443 (2007). [CrossRef]
- M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998). [CrossRef]
- S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007). [CrossRef]
- M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1986).

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