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

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References

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  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]
  2. H. Xu, “Electromagnetic energy flow near nanoparticles.I: single spheres,” J. Quant. Spectrosc. Radiat. Transfer 87, 53–67 (2004).
    [CrossRef]
  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. 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]
  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. 16, 1198–1200 (1993).
  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 82, 036703-1–036703-12 (2010).
    [CrossRef]
  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: (2011).
    [CrossRef]
  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]
  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,  56, 141–163 (1996).
    [CrossRef]

2011

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]

2010

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]

2008

2004

H. Xu, “Electromagnetic energy flow near nanoparticles.I: single spheres,” J. Quant. Spectrosc. Radiat. Transfer 87, 53–67 (2004).
[CrossRef]

1998

1996

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

1994

1993

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).

1991

Barber, P. W.

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

da Cunha, R.

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).

Draine, B. T.

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]

B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 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).

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]

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

Flatau, P. J.

Goodman, J. J.

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).

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]

Gutkowicz-Krusin, D.

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

Hergert, W.

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]

Hill, S. C.

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

Hoekstra, A. G.

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]

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]

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]

Hopkins, T.

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).

Karamehmedovic, M.

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]

Matyssek, C.

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]

Min, M.

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]

Rahola, J.

Schmidt, V.

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]

Schuh, R.

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]

Seifert, G.

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]

Sleijpen, G.

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

Sloot, P.

Stalmashonak, A.

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]

Stranik, O.

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]

van der Vorst, H.

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

Wriedt, T.

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]

Xu, H.

H. Xu, “Electromagnetic energy flow near nanoparticles.I: single spheres,” J. Quant. Spectrosc. Radiat. Transfer 87, 53–67 (2004).
[CrossRef]

Yurkin, M. A.

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]

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]

Appl. Opt.

Astrophys. J.

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).

Computing

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

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

H. Xu, “Electromagnetic energy flow near nanoparticles.I: single spheres,” J. Quant. Spectrosc. Radiat. Transfer 87, 53–67 (2004).
[CrossRef]

Opt. Exp.

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]

Opt. Lett.

Phys. Rev. E

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]

Other

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]

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).

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

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

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Figures (2)

Fig. 1
Fig. 1

Normalized electric field intensity |E|/|E0| inside and in the proximity of a sphere calculated on a 60×60×60 grid, on a plane passing through the center of the sphere. The sphere is modeled by 33552 dipoles on a grid with d = D/40. The refractive index m = 0.96 + 1.01i and x = k0D/2 = 5 (e.g., D = 796nm Au sphere and λ = 500nm). For this highly absorptive case E is small inside the sphere. The incoming light is linearly polarized with Einc || ŷ, and propagating in the + direction.

Fig. 2
Fig. 2

Computational time required for problem of Fig. 1, with E calculated in a 1.5D×1.5D×1.5D volume centered on the sphere. Method 1 calculations were done with ddfield from the DDSCAT 7.1 distribution. Method 2 calculations were done with m = 1.000001+0i for the pseudovacuum. Method 3 calculations were done with DDSCAT 7.2.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

E inc ( r , t ) = E 0 exp ( i k 0 r i ω t ) ,
E i = E inc , i j i A ˜ i j P j
P i = α i [ E inc ( r i ) j i A ˜ i j P j ] ,
E inc , i = j target A i j P j .
E i = E inc , i j target A ˜ i j P j ,
E i ( at dipole sites ) = α i 1 P i ,
m i = { m target original target sites m vacuum 1 for the pseudo vacuum sites .
P i ( extended target ) = { P target original target sites P vacuum pseudo vacuum sites .
E = E inc + E scat = { α j 1 P j original target sites j E inc , i j target A ˜ i j P j vacuum sites i .
P i ( extended target ) = { P target original target sites 0 vacuum sites .

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