## Fast near field calculations in the discrete dipole approximation for regular rectilinear grids |

Optics Express, Vol. 20, Issue 2, pp. 1247-1252 (2012)

http://dx.doi.org/10.1364/OE.20.001247

Acrobat PDF (692 KB)

### Abstract

A near-field calculation of light electric field intensity inside and in the vicinity of a scattering particle is discussed in the discrete dipole approximation. A fast algorithm is presented for gridded data. This algorithm is based on one matrix times vector multiplication performed with the three dimensional fast Fourier transform. It is shown that for moderate and large light scattering near field calculations the computer time required is reduced in comparison to some of the other methods.

© 2011 OSA

## 1. Methods

### 1.1. Introduction

1. A. G. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. **37**, 8482–8497 (1998). [CrossRef]

4. M. Karamehmedovic, R. Schuh, V. Schmidt, T. Wriedt, C. Matyssek, W. Hergert, A. Stalmashonak, G. Seifert, and O. Stranik, “Comparison of numerical methods in near-field computation for metallic nanoparticles,” Opt. Exp. **19**, 8939–8953 (2011). [CrossRef]

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

*α*are specified, Maxwell’s equations can be solved accurately for the dipole array. For a monochromatic incident plane wave the polarizations

_{j}**P**

*of the dipoles in the target oscillate coherently. Each dipole*

_{j}*i*is affected by the incident wave plus the electric field at location

*i*due to all the other point dipoles (see [5

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

**Ã**

*). The vector of polarizations*

_{ij}**P**

*must satisfy the system of equations or, defining*

_{j}*N*dipoles, Eq. (3) is a system of 3

*N*linear equations. The polarizability tensors

*α*are obtained from lattice dispersion relation theory [5

_{j}5. B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**, 1491–1499 (1994). [CrossRef]

8. M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: Filtered coupled dipoles revived,” Phys. Rev. E **82**, 036703-1–036703-12 (2010). [CrossRef]

**Ã**depends only on the ratio of interdipole separation to wavelength. After

**Ã**has been calculated, Eq. (4) can be solved for

**P**

*, using iterative techniques when*

_{j}*N*≫ 1.

### 1.2. Method 1

**P**

*are obtained from Eq. (4) the electric field at location*

_{j}*i*can be calculated from where

**P**

*are now known dipole polarizations. The interaction matrix*

_{j}**Ã**

*is spatially invariant and depends only on the displacement between*

_{ij}*i*and

*j*. For the special case of near field calculation at the dipole locations inside the target, instead of performing sums in Eq. (5) one can immediately obtain the electric field

**E**=

**E**

_{inc}+

**E**

_{scat}as where the 3×3 tensor

*α*depends only on the local refractive index

_{i}*m*. In many applications one is often interested in the near field outside the target. This can be evaluated directly from Eq. (5); this method was implemented in program ddfield in version 7 of DDSCAT [9

9. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A **25**, 2693–2703 (2008). [CrossRef]

10. M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer, doi: [CrossRef] (2011).

*N*

_{loc}is large this method of direct summmation, with complexity

*O*(3

*N*

_{target}× 3

*N*

_{loc}), is computationally very slow.

### 1.3. Method 2

*m*is set close to 1 for the vacuum sites and solving scattering problem Eq. (3) directly. In such case This gives

**E**at grid sites is obtained directly from Eq. (6). The problem with such an approach is that the iterative procedure is carried out for an extended target and that (for example, in the current DDSCAT implementation) the refractive index of pseudo-vacuum sites can not be set exactly to 1 as

11. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of FFT Techniques to the Discrete Dipole Approximation,” Opt. Lett. **16**, 1198–1200 (1991). [CrossRef] [PubMed]

*O*[3

*N*

_{target+vacuum}×

*M*log(3

*N*

_{target+vacuum})], where

*M*is number of conjugate gradient iterations. This can be much faster than method 1, but is burdened by the need to include the pseudo-vacuum sites in the iterative solution of Eq. (4). This method is also mentioned in documentation of the ADDA code [8

8. M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: Filtered coupled dipoles revived,” Phys. Rev. E **82**, 036703-1–036703-12 (2010). [CrossRef]

*d*be small enough that the structure in both

**E**and the target be well-resolved (the usual criterion is |

*m*|

*k*

_{0}

*d*< 0.5) and therefore this grid is well-suited for sampling the near-field.

## 2. Method 3 - main results

**E**on a user-selected grid one first does a DDA calculation for the target shape with

*N*

_{target}dipole sites to obtain the polarizations vector

**P**

_{target}. Then, using the original lattice on which the dipoles are located, we define a rectangular volume within which we wish to compute

**E**. The matrix times vector calculation in Eq. (5) is performed, giving the solution The algorithmic complexity is

*O*[3

*N*

_{target+sites}× log(3

*N*

_{target+sites})]. This method exploits the fact that calculation of Eq. (5) can be performed efficiently using FFTs [11

11. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of FFT Techniques to the Discrete Dipole Approximation,” Opt. Lett. **16**, 1198–1200 (1991). [CrossRef] [PubMed]

## 3. Implementation and results

**Ã**is not explicitly stored. We perform two calculations using DDSCAT. First we solve for the polarization

**P**

*at the original target dipole sites. The second calculation extends the original target shape with the vacuum sites with*

_{j}**P**

*= 0 at which*

_{i}**E**is to be calculated. This gives the following extended polarization vector This extended box-like shape forms a rectilinear grid with the total number of dipoles

*N*

_{ext}. The grid contains all of the original sites and extended vacuum sites.

*D*= 40

*d*the results are presented in Fig. 1. The total internal and near field electric field intensity is shown in a rectangular volume of size 1.5

*D*×1.5

*D*×1.5

*D*(60×60×60 lattice sites) centered on the sphere. The refractive index is

*m*= 0.96 + 1.01

*i*and size parameter

*x*= 5. For this highly absorptive case the intensity of the total electric field is small inside the sphere.

^{−5}. In Method 2, the pseudovacuum refractive index was set to 1 + 10

^{−6}. For Methods 1 and 3 we used the PBCGST method from the Parallel Iterative Methods (PIM) package [12], which converged in 11–12 iterations. For Method 2, the PBCGST method failed to converge because of numerical problems associated with the small polarizabiities of the pseudovacuum sites, and we instead used the PBCGS2 method [13

13. G. Sleijpen and H. van der Vorst, “Reliable updated residuals in hybrid BiCG methods,” Computing , **56**, 141–163 (1996). [CrossRef]

*N*

_{target}×

*N*

_{nearfield}scaling for Method 1 vs. the

*N*log

*N*scaling of Method 3, where

*N*

_{target}is the number of dipoles in the target,

*N*

_{nearfield}is the number of points where the nearfield is calculated, and

*N*=

*N*

_{target}+

*N*

_{nearfield}.

*N*

_{nearfield}of pseudo-vacuum dipoles needed for the extended problem, the value of the refractive index used for the pseudo-vacuum dipole sites, and the conjugate gradient method used. In the limiting case of

*no*additional pseudo-vacuum dipoles, i.e. when near field is only needed inside the target, the numerical complexity of Methods 2 and 3 is the same. In the limit of very large number of pseudo-vacuum sites in comparison to occupied target sites the time difference will be dominated by the matrix times vector multiplications. Such difference depends on number of iterations and number of matrix times vector multiplications needed per conjugate gradient method iteration (typically between 6–12). In more realistic cases of similar numbers of target sites and pseudo-vacuum sites, like the one presented here (with

*N*

_{nearfield}/

*N*

_{target}≈ (81/4

*π*) ≈ 6.5) the total time for method 2 and 3 is dominated by the solution time itself, i.e. it is proportional to

*MN*log

*N*. However, at least in the DDSCAT implementation, the accuracy of method 2 is limited because the solution diverges for refractive index equal to vacuum.

**P**

_{target}, and the additional time required for the nearfield calculation after

**P**

_{target}has been found. For this case, the time required to iteratively solve for

**P**

_{target}is about 5 times greater than the time required by the subsequent nearfield calculation.

## 4. Summary

## Acknowledgments

## References and links

1. | A. G. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. |

2. | H. Xu, “Electromagnetic energy flow near nanoparticles.I: single spheres,” J. Quant. Spectrosc. Radiat. Transfer |

3. | P. W. Barber and S. C. Hill, “Light scattering by particles: computational methods,” World Scientific Publishing, Singapore, ISBN:9971-50-813-3 (1990). |

4. | M. Karamehmedovic, R. Schuh, V. Schmidt, T. Wriedt, C. Matyssek, W. Hergert, A. Stalmashonak, G. Seifert, and O. Stranik, “Comparison of numerical methods in near-field computation for metallic nanoparticles,” Opt. Exp. |

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

6. | B. T. Draine and J. J. Goodman, “Beyond Clausius-Mossotti: Wave Propagation on a Polarizable Point Lattice and the Discrete Dipole Approximation,” Astrophys. J. |

7. | D. Gutkowicz-Krusin and B. T. Draine, “Propagation of Electromagnetic Waves on a Rectangular Lattice of Polarizable Points,” arXiv:astro-ph/0403082 (2004). |

8. | M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: Filtered coupled dipoles revived,” Phys. Rev. E |

9. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A |

10. | M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer, doi: [CrossRef] (2011). |

11. | J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of FFT Techniques to the Discrete Dipole Approximation,” Opt. Lett. |

12. | R. da Cunha and T. Hopkins, “PIM 2.0 The Parallel Iterative Methods Package for Systems of Linear Equations User’s Guide (Fortran 77 version),” Technical report. UKC, University of Kent, Canterbury, UK (1996). |

13. | G. Sleijpen and H. van der Vorst, “Reliable updated residuals in hybrid BiCG methods,” Computing , |

**OCIS Codes**

(290.0290) Scattering : Scattering

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: October 27, 2011

Manuscript Accepted: November 2, 2011

Published: January 5, 2012

**Citation**

P. J. Flatau and B. T. Draine, "Fast near field calculations in the discrete dipole approximation for regular rectilinear grids," Opt. Express **20**, 1247-1252 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1247

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### References

- A. G. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt.37, 8482–8497 (1998). [CrossRef]
- H. Xu, “Electromagnetic energy flow near nanoparticles.I: single spheres,” J. Quant. Spectrosc. Radiat. Transfer87, 53–67 (2004). [CrossRef]
- P. W. Barber and S. C. Hill, “Light scattering by particles: computational methods,” World Scientific Publishing, Singapore, ISBN:9971-50-813-3 (1990).
- M. Karamehmedovic, R. Schuh, V. Schmidt, T. Wriedt, C. Matyssek, W. Hergert, A. Stalmashonak, G. Seifert, and O. Stranik, “Comparison of numerical methods in near-field computation for metallic nanoparticles,” Opt. Exp.19, 8939–8953 (2011). [CrossRef]
- B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A11, 1491–1499 (1994). [CrossRef]
- B. T. Draine and J. J. Goodman, “Beyond Clausius-Mossotti: Wave Propagation on a Polarizable Point Lattice and the Discrete Dipole Approximation,” Astrophys. J.16, 1198–1200 (1993).
- D. Gutkowicz-Krusin and B. T. Draine, “Propagation of Electromagnetic Waves on a Rectangular Lattice of Polarizable Points,” arXiv:astro-ph/0403082 (2004).
- M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: Filtered coupled dipoles revived,” Phys. Rev. E82, 036703-1–036703-12 (2010). [CrossRef]
- B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A25, 2693–2703 (2008). [CrossRef]
- M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer, doi: (2011). [CrossRef]
- J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of FFT Techniques to the Discrete Dipole Approximation,” Opt. Lett.16, 1198–1200 (1991). [CrossRef] [PubMed]
- R. da Cunha and T. Hopkins, “PIM 2.0 The Parallel Iterative Methods Package for Systems of Linear Equations User’s Guide (Fortran 77 version),” Technical report. UKC, University of Kent, Canterbury, UK (1996).
- G. Sleijpen and H. van der Vorst, “Reliable updated residuals in hybrid BiCG methods,” Computing, 56, 141–163 (1996). [CrossRef]

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