## Comparison of numerical methods in near-field computation for metallic nanoparticles |

Optics Express, Vol. 19, Issue 9, pp. 8939-8953 (2011)

http://dx.doi.org/10.1364/OE.19.008939

Acrobat PDF (1391 KB)

### Abstract

Four widely used electromagnetic field solvers are applied to the problem of scattering by a spherical or spheroidal silver nanoparticle in glass. The solvers are tested in a frequency range where the imaginary part of the scatterer refractive index is relatively large. The scattering efficiencies and near-field results obtained by the different methods are compared to each other, as well as to recent experiments on laser-induced shape transformation of silver nanoparticles in glass.

© 2011 OSA

## 1. Introduction

1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys.-Leipzig **330**, 377–445 (1908). [CrossRef]

3. J. Niegemann, M. Konig, K. Stannigel, and K. Busch, “Higher order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostructures **7**(1), 2–11 (2009). [CrossRef]

5. A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by stair-stepped approximation of a conduction boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antenn. Propag. **39**(10), 1518–1525 (1991). [CrossRef]

7. A. Vial and T. Laroche, “Description of dispersion properties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method,” J. Phys. D Appl. Phys. **40**(22), 7152–7158 (2007). [CrossRef]

9. A. Doicu and T. Wriedt, “Near-field computation using the null-field method,” J. Quant. Spectrosc. Radiat. Transf. **111**(3), 466–473 (2010). [CrossRef]

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

11. B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT 7.1”, http://arXiv.org/abs/1002.1505v1 (2010).

12. COMSOL Multiphysics demonstration CD-ROM can be requested at http://www.comsol.com

## 2. The test scattering problem

*x*/

*a*)

^{2}+ (

*y*/

*a*)

^{2}+ (

*z*/

*b*)

^{2}= 1, with

*a*=

*b*= 15nm in the first case and

*a*≈11.2nm,

*b*≈26.9nm in the second case. The scatterer is immersed in glass with refractive index 1.5, and illuminated by a time-harmonic,

*z-*polarized, uniform plane wave of amplitude 1 V/m and propagating in the positive

*x*-direction. The free-space wavelength λ

_{0}of the incident field is allowed in the range 250nm–900nm. Figure 1a shows the real and the imaginary part of the refractive index

*n*

_{Ag}of silver for these wavelengths, obtained by spline interpolation of the data from Lynch and Hunter [6].

## 3. Laser-induced shaping of silver nanoparticles

15. A. Stalmashonak, G. Seifert, and H. Graener, “Spectral range extension of laser-induced dichroism in composite glass with silver nanoparticles,” J. Opt. A, Pure Appl. Opt. **11**(6), 065001 (2009). [CrossRef]

16. A. Stalmashonak, C. Matyssek, O. Kiriyenko, W. Hergert, H. Graener, and G. Seifert, “Preparing large-aspect-ratio prolate metal nanoparticles in glass by simultaneous femtosecond multicolor irradiation,” Opt. Lett. **35**(10), 1671–1673 (2010). [CrossRef] [PubMed]

14. A. Stalmashonak, A. Podlipensky, G. Seifert, and H. Graener, “Intensity-driven, laser induced transformation of Ag nanospheres to anisotropic shapes,” Appl. Phys. B **94**(3), 459–465 (2009). [CrossRef]

## 4. Numerical methods

9. A. Doicu and T. Wriedt, “Near-field computation using the null-field method,” J. Quant. Spectrosc. Radiat. Transf. **111**(3), 466–473 (2010). [CrossRef]

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

11. B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT 7.1”, http://arXiv.org/abs/1002.1505v1 (2010).

12. COMSOL Multiphysics demonstration CD-ROM can be requested at http://www.comsol.com

17. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. **19**(9), 1505–1509 (1980). [CrossRef] [PubMed]

*N*of 3 or 4. For non-spherical particles of the same volume, or when the electromagnetic field needs to be evaluated close to the scatterer surface, the number of multipoles should be increased for accuracy. We use the maximal order

*N*= 8 of SVWF in the field expressions. For the near-field data of Section 5, the radiating integrals are computed by discretizing the generatrix of the particle surface by 25 up to 400 points.

*N*polarizable point dipoles. The dipoles are placed at points

**r**

*in a cubic lattice inside the particle volume and with lattice spacing*

_{μ}*d*. For isotropic materials, the polarization vector

**r**

*is assumed equal to*

_{μ}**r**

*) of the incident electric field and the properly retarded electric fields radiated by all the other dipoles in the array. It holds with the time dependence*

_{μ}18. D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points”, http://xxx.arxiv.org/abs/astro-ph/0403082 (2004).

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

11. B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT 7.1”, http://arXiv.org/abs/1002.1505v1 (2010).

*d*be small compared to all structural lengths of the particle (so that the dipole array models the shape of the particle with sufficient accuracy), and that

*d*be sufficiently small compared to the plane-wave wavelength in the particle material. The second criterion may be satisfied [10

**11**(4), 1491–1499 (1994). [CrossRef]

*n*is the complex refractive index of the particle material and

*d*= 1.5nm. The structural lengths of the considered particles are thus at least 20 times larger than the used interdipole separation, and, since

^{−5}. Finally, MMP 3D is a semi-analytic code based on the Multiple Multipole (MMP) expansion. This solver is applicable to scattering problems involving piecewise linear, homogeneous and isotropic domains. In MMP 3D, the fields are approximated by linear combinations of elementary solutions of the Maxwell system, such as plane waves, waveguide modes and 2D and 3D multipoles. The availability of different types of elementary solutions allows efficient formulation of practical problems, tailored to the given geometry. In a typical scattering formulation, the scattered field and the field in the interior of the scatterer are expressed in terms of sums of multipoles with different origins of expansion,

*j*) signifies the

*j*’th origin of expansion.) The coefficients in the sums are found by generalized point matching at the scatterer boundary. The number of testing points where the boundary condition is enforced is usually much larger than the number of used multipoles. For the numerical results presented below, the boundary condition is tested at 255 points on the particle surface, and a total of

*n*= 6 multipole solutions are used in the representation of

*z*-axis in the interior of the particle. The symmetry of the scattering problem with respect to the

*xy*-plane is exploited to halve the actual number of unknown coefficients. The radiating and regular multipoles have coinciding origins of expansion, and they employ SVWFs of maximal order 20, 11 and 6, and maximal degree 7, 11 and 6, respectively. The maximal order of the SVWFs could be reduced without a significant increase in the solution error, but, as seen in Section 5, the near-field execution time for MMP 3D is relatively short even for the above choice of the number of SVWFs used.

## 5. Numerical results

*is done for the solver parameters fixed in*Section 4. The execution times stated below are included for illustration only, since the solvers were run on different computers. The NFM-DS was run on a single core of a 2.33GHz Intel® Xeon® processor with 16Gb RAM, DDSCAT and MMP 3D used a single core of a 2.66GHz Intel® Core2 Duo CPU with 4Gb RAM, and COMSOL was run on a 2.4GHz Intel® Core2 Quad processor with 8Gb RAM. For the near-field comparisons, the free-space incident wavelengths λ

_{0}= 410nm and λ

_{0}= 688nm are used, since they correspond to off-resonance and on-resonance values, respectively. The computed total electric near fields are sampled on subsets of a fixed rectangular grid

*G*in the

*xz*-plane, i.e., in the plane spanned by the direction of propagation of the incident field and by the incident polarization. The 121 × 86 grid consists of points (

*x*,

_{j}*z*) spaced equidistantly between −60nm and 60nm along the

_{j}*x*-axis and between 0nm and 85nm along the

*z*-axis (the spacing is 1nm). Since, as it turns out, the quality of the computed near field can deteriorate significantly as the observation point approaches the NP surface, it is convenient to have a measure of discrepancy between the near fields that disregards the field values in a well-defined, variable neighborhood of the NP. Thus, for the spherical scatterer the

**E**is in the following defined byand the corresponding

**E**

_{1}and

**E**

_{2}is defined as

*xz*-plane. Similarly, for the prolate spheroidal scatterer the

**E**is defined for 1 < ε < 3 byand the corresponding

**E**

_{1}and

**E**

_{2}is defined as

*xz*-plane. In both cases, the closer the parameter ε is to 1 the more the field in the immediate vicinity of the scatterer is taken into account when calculating the field norms and distances. Figure 2 shows the relative

_{0}= 410nm, where we have chosen the semi-analytic MMP as reference solution. (MMP 3D is chosen as reference because of the absence of discretized surface integrals in that method. Analytic expressions for scattering by spheroidal particles exist, but, in contrast to the spherical case, the evaluation of the near field is not easily available.) The near-field error is plotted on a logarithmic scale. In NFM-DS, 400 discretization points are used along the generatrix of the particle surface, and the DDSCAT results are shown for arrays of 20 × 20 × 20 and 40 × 40 × 40 point dipoles modeling the particle.

*G*outside the scatterer are taken into account. Next, Fig. 3 shows the scattering efficiency of the prolate spheroidal scatterer as function of the incident free-space wavelength (from 250nm to 900nm), as calculated by all four solvers. NFM-DS and COMSOL agree well with the reference MMP 3D solution over almost the entire shown wavelength spectrum. There is very good agreement between the DDSCAT and MMP results in the range 250nm–400nm. However, DDSCAT with a 20 × 20 × 48 array shows a relatively large error in the range 550nm–650nm, and it overestimates the scattering efficiency for longer wavelengths. It turns out that modeling the scatterer with an array of 40 × 40 × 96 points shows only a relatively small improvement in the computed scattering efficiency, and that the number of polarisable points needs to be much larger (e.g., 80 × 80 × 192, corresponding to the lattice spacing

*d =*0.28nm and

_{0}= 318nm and λ

_{0}= 688nm, respectively. Again, the near-field error is relative to the MMP 3D result, and it is plotted on a logarithmic scale. For NFM-DS, results with 50 and 400 generatrix discretization points are shown, and for DDSCAT results with arrays of 20 × 20 × 48 and 40 × 40 × 96 point dipoles are given. For the shown values of ε, and for methods other than NFM-DS 50, the relative near-field error is less than 10% at 318nm and less than 18% at 688nm. For large ε, at λ

_{0}= 318nm, the relative error seems to settle at 10

^{−6}for NFM-DS 50 and NFM-DS 400, at 10

^{−5}for DDSCAT 40 × 40 × 96 and at 10

^{−3}for COMSOL. At λ

_{0}= 688nm, the relative error seems to settle at 10

^{-4.5}≈0.003% for NFM-DS 50 and NFM-DS 400, at 10

^{−2}for DDSCAT 40 × 40 × 96 and at 10

^{-1.65}≈2.2% for COMSOL. The difference between the error levels in Figs. 4 and 5 is likely due to the fact that

*n*

_{Ag}= 0.93 +

*i*0.51 (with time dependence

*n*

_{Ag}= 0.14 +

*i*4.44 at 688nm, that is, Im

*n*

_{Ag}≈0.55 Re

*n*

_{Ag}at 318nm while Im

*n*

_{Ag}≈32 Re

*n*

_{Ag}at 688nm. The quality of the near field decreases significantly for NFM-DS, DDSCAT and COMSOL as ε approaches 1. While NFM-DS produces by far the smallest error overall, COMSOL appears to give the best-quality near field closest to the scatterer boundary. For ε close to 1, the relative near-field error produced by NFM-DS 400 is approximately 3% (worse than DDSCAT) at 318nm and approximately 10% (better than DDSCAT) at 688nm. In contrast, the corresponding COMSOL error is approximately 0.2% at 318nm and approximately 8% at 688nm. Note that, for both incident wavelengths, the COMSOL relative error varies the least with respect to the parameter ε. This, as well as the relatively good quality of the COMSOL near field close to the particle, could be due to the fact that the COMSOL mesh density increases as the distance to the particle decreases, while the other methods rely on radiating sources at the particle surface or inside the particle volume and do not compensate for the proximity of the observation point to the particle. Figures 6 and 7 show the relative near-field error for NFM-DS with 25 up to 400 generatrix discretization points, for λ

_{0}= 318nm and λ

_{0}= 688nm, respectively. As expected, in both cases the near-field quality increases with better discretization. The calculation of the near field over the grid

*G*took (for both incident wavelengths) approx. 16 sec for NFM-DS 25, 31 sec for NFM-DS 50, 60 sec for NFM-DS 100, 115 sec for NFM-DS 200 and 238 sec for NFM-DS 400. For DDSCAT 40 × 40 × 96, this calculation took approx. 297 sec at 318nm and 619 sec at 688nm. COMSOL took 126 sec at 318 nm and 156 sec at 688nm, and MMP 3D took 6 sec. The COMSOL times include mesh generation and assembly.

*xz*-plane (see the coordinate system shown in Fig. 1b) obtained at λ

_{0}= 318nm by the NFM-DS, DDSCAT and MMP 3D, respectively. Figure 9 shows these data in the

*yz*-plane. The near field is evaluated on an 0.5nm rectangular mesh with points spaced equidistantly along the

*x*and z axes. At 318nm incident wavelength,

*n*

_{Ag}= 0.93 +

*i*0.51 and the imaginary part of

*n*

_{Ag}attains a global minimum in the chosen wavelength range (see Fig. 1a). As attested by the above analysis in terms of

*exterior to the scatterer*are in good agreement; in particular, they show local minima near the poles of the prolate spheroid. The

*interior*fields of NFM-DS, COMSOL and MMP agree, while the interior field produced by DDSCAT has a significantly lower amplitude and appears more homogeneous in comparison. Figures 10 and 11 show the near-field results in the

*xz*-plane and the

*yz*-plane, respectively, obtained at λ

_{0}= 688nm incident wavelength. Again, the log

_{10}-values of the total near-field amplitudes are shown. For the incident wavelength of 688nm,

*n*

_{Ag}= 0.14 +

*i*4.44 and the imaginary part of

*n*

_{Ag}is substantial – about 32 times larger than the real part. The exterior fields are in good agreement, with significant field enhancement near the poles of the scatterer and local minima in its equatorial plane. NFM-DS, COMSOL and MMP 3D agree in the interior of the scatterer, while the DDSCAT field shows unphysical features there.

## 6. Conclusion

## Acknowledgments

## References and links

1. | G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys.-Leipzig |

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

3. | J. Niegemann, M. Konig, K. Stannigel, and K. Busch, “Higher order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostructures |

4. | P. Monk, |

5. | A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by stair-stepped approximation of a conduction boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antenn. Propag. |

6. | D. W. Lynch and W. R. Hunter, “Comments on the Optical Constants of Metals and an Introduction to the Data for Several Metals,” in |

7. | A. Vial and T. Laroche, “Description of dispersion properties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method,” J. Phys. D Appl. Phys. |

8. | A. Doicu, T. Wriedt and Yu. Eremin, |

9. | A. Doicu and T. Wriedt, “Near-field computation using the null-field method,” J. Quant. Spectrosc. Radiat. Transf. |

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

11. | B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT 7.1”, http://arXiv.org/abs/1002.1505v1 (2010). |

12. | COMSOL Multiphysics demonstration CD-ROM can be requested at http://www.comsol.com |

13. | C. Hafner and L. Bomholt, |

14. | A. Stalmashonak, A. Podlipensky, G. Seifert, and H. Graener, “Intensity-driven, laser induced transformation of Ag nanospheres to anisotropic shapes,” Appl. Phys. B |

15. | A. Stalmashonak, G. Seifert, and H. Graener, “Spectral range extension of laser-induced dichroism in composite glass with silver nanoparticles,” J. Opt. A, Pure Appl. Opt. |

16. | A. Stalmashonak, C. Matyssek, O. Kiriyenko, W. Hergert, H. Graener, and G. Seifert, “Preparing large-aspect-ratio prolate metal nanoparticles in glass by simultaneous femtosecond multicolor irradiation,” Opt. Lett. |

17. | W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. |

18. | D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points”, http://xxx.arxiv.org/abs/astro-ph/0403082 (2004). |

19. | M. A. Yurkin, “Discrete dipole simulations of light scattering by blood cells”, Dissertation (2007), ISBN 90–5776–169–6 |

**OCIS Codes**

(290.0290) Scattering : Scattering

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 4, 2011

Revised Manuscript: March 11, 2011

Manuscript Accepted: March 16, 2011

Published: April 22, 2011

**Citation**

Mirza Karamehmedović, Roman Schuh, Vladimir Schmidt, Thomas Wriedt, Christian Matyssek, Wolfram Hergert, Andrei Stalmashonak, Gerhard Seifert, and Ondrej Stranik, "Comparison of numerical methods in near-field computation for metallic nanoparticles," Opt. Express **19**, 8939-8953 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8939

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### References

- G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys.-Leipzig 330, 377–445 (1908). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics, Third Edition. (Artech House 2005).
- J. Niegemann, M. Konig, K. Stannigel, and K. Busch, “Higher order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostructures 7(1), 2–11 (2009). [CrossRef]
- P. Monk, Finite Element Method for Maxwell’s Equations, (Oxford, 2006).
- A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by stair-stepped approximation of a conduction boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antenn. Propag. 39(10), 1518–1525 (1991). [CrossRef]
- D. W. Lynch and W. R. Hunter, “Comments on the Optical Constants of Metals and an Introduction to the Data for Several Metals,” in Handbook of Optical Constants of Solids, vol. 1, E. D. Palik, ed (Academic, San Diego, 1985).
- A. Vial and T. Laroche, “Description of dispersion properties of metals by means of the critical points model and application to the study of resonant structures using the FDTD method,” J. Phys. D Appl. Phys. 40(22), 7152–7158 (2007). [CrossRef]
- A. Doicu, T. Wriedt and Yu. Eremin, Light Scattering by Systems of Particles, Null-Field Method with Discrete Sources: Theory and Programs (Springer 2006).
- A. Doicu and T. Wriedt, “Near-field computation using the null-field method,” J. Quant. Spectrosc. Radiat. Transf. 111(3), 466–473 (2010). [CrossRef]
- B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]
- B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT 7.1”, http://arXiv.org/abs/1002.1505v1 (2010).
- COMSOL Multiphysics demonstration CD-ROM can be requested at http://www.comsol.com
- C. Hafner and L. Bomholt, The 3D electromagnetic wave simulator (Wiley 1993).
- A. Stalmashonak, A. Podlipensky, G. Seifert, and H. Graener, “Intensity-driven, laser induced transformation of Ag nanospheres to anisotropic shapes,” Appl. Phys. B 94(3), 459–465 (2009). [CrossRef]
- A. Stalmashonak, G. Seifert, and H. Graener, “Spectral range extension of laser-induced dichroism in composite glass with silver nanoparticles,” J. Opt. A, Pure Appl. Opt. 11(6), 065001 (2009). [CrossRef]
- A. Stalmashonak, C. Matyssek, O. Kiriyenko, W. Hergert, H. Graener, and G. Seifert, “Preparing large-aspect-ratio prolate metal nanoparticles in glass by simultaneous femtosecond multicolor irradiation,” Opt. Lett. 35(10), 1671–1673 (2010). [CrossRef] [PubMed]
- W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19(9), 1505–1509 (1980). [CrossRef] [PubMed]
- D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points”, http://xxx.arxiv.org/abs/astro-ph/0403082 (2004).
- M. A. Yurkin, “Discrete dipole simulations of light scattering by blood cells”, Dissertation (2007), ISBN 90–5776–169–6

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