Abstract

During the development and application of a scattering algorithm, its accuracy is normally validated by comparing with results of spherical particles given by the exact Mie theory. Being the simplest shape, sphere supports morphology-dependent resonances (MDRs), which cause sharp variations of the scattering properties in narrow size ranges. We show that MDRs may mislead the validation of any volume- or surface-discretization methods, including the discrete dipole approximation (DDA) and, thus, should be explicitly avoided. However, the brute-force DDA simulations can actually capture the narrow peaks in the extinction efficiency over the size parameter, but only if a dipole size parameter is smaller than twice the MDR width. That is much more computationally intensive than typical DDA simulations. We find that a single Lorentzian MDR peak may be split into two due to the symmetry breaking by the DDA discretization. Furthermore, instead of time-consuming high-resolution DDA simulations for reproducing MDR, we developed and validated a significantly more computationally efficient method. It is based, first, on fitting simulated data with one or two Lorentzian peaks combined with a cubic baseline. Second, we use Richardson extrapolation of peak parameters to zero dipole size, exploiting the smooth convergence of these parameters towards the reference Mie values. When applied to two MDRs with relative widths 2 × 10−3 and 9 × 10−4, the developed workflow, powered by intensive simulations, reproduces the peak positions with unprecedented accuracy – errors less than 0.07% and 0.4% of their widths, respectively. This extends the way for studying the evolution of the MDR under non-axisymmetric deformations of a sphere or a spheroid.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  33. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A 23(10), 2578–2591 (2006).
    [Crossref] [PubMed]
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    [Crossref]

2018 (1)

C. Liu, S. Teng, Y. Zhu, M. A. Yurkin, and Y. L. Yung, “Performance of the discrete dipole approximation for optical properties of black carbon aggregates,” J. Quant. Spectrosc. Radiat. Transf. 221, 98–109 (2018).
[Crossref]

2017 (1)

A. Moridnejad, T. C. Preston, and U. K. Krieger, “Tracking water sorption in glassy aerosol particles using morphology-dependent resonances,” J. Phys. Chem. A 121(42), 8176–8184 (2017).
[Crossref] [PubMed]

2015 (1)

D. A. Smunev, P. C. Chaumet, and M. A. Yurkin, “Rectangular dipoles in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transf. 156, 67–79 (2015).
[Crossref]

2014 (1)

2013 (1)

M. A. Yurkin, “Symmetry relations for the Mueller scattering matrix integrated over the azimuthal angle,” J. Quant. Spectrosc. Radiat. Transf. 131, 82–87 (2013).
[Crossref]

2012 (1)

2011 (1)

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

2010 (1)

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 Stat. Nonlin. Soft Matter Phys. 82(3 Pt 2), 036703 (2010).
[Crossref] [PubMed]

2007 (3)

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 558–589 (2007).
[Crossref]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 546–557 (2007).
[Crossref]

2006 (3)

2003 (1)

2000 (1)

M. I. Mishchenko and A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116(1–2), 167–179 (2000).
[Crossref]

1998 (2)

A. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. 37(36), 8482–8497 (1998).
[Crossref] [PubMed]

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. I. Sindoni, “Optical resonances of spheres containing an eccentric spherical inclusion Résonances optiques de sphères contenant une inclusion sphérique excentrée,” J. Opt. 29(1), 28–34 (1998).
[Crossref]

1995 (1)

1994 (2)

1993 (1)

1992 (2)

1990 (1)

1984 (1)

1978 (1)

P. Chýlek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18(5), 2229–2233 (1978).
[Crossref]

1977 (1)

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38(23), 1351–1354 (1977).
[Crossref]

1976 (1)

1968 (1)

1909 (1)

P. Debye, “Der lichtdruck auf kugeln von beliebigem material,” Ann. Phys. 335(11), 57–136 (1909).
[Crossref]

Ashkin, A.

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38(23), 1351–1354 (1977).
[Crossref]

Barber, P. W.

Bi, L.

Borghese, F.

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. I. Sindoni, “Optical resonances of spheres containing an eccentric spherical inclusion Résonances optiques de sphères contenant une inclusion sphérique excentrée,” J. Opt. 29(1), 28–34 (1998).
[Crossref]

Budko, N. V.

N. V. Budko and A. B. Samokhin, “Spectrum of the volume integral operator of electromagnetic scattering,” SIAM J. Sci. Comput. 28(2), 682–700 (2006).
[Crossref]

Chaumet, P. C.

D. A. Smunev, P. C. Chaumet, and M. A. Yurkin, “Rectangular dipoles in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transf. 156, 67–79 (2015).
[Crossref]

Chýlek, P.

Conwell, P. R.

Debye, P.

P. Debye, “Der lichtdruck auf kugeln von beliebigem material,” Ann. Phys. 335(11), 57–136 (1909).
[Crossref]

Denti, P.

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. I. Sindoni, “Optical resonances of spheres containing an eccentric spherical inclusion Résonances optiques de sphères contenant une inclusion sphérique excentrée,” J. Opt. 29(1), 28–34 (1998).
[Crossref]

Draine, B. T.

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[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]

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38(23), 1351–1354 (1977).
[Crossref]

Flatau, P. J.

Fuchs, R.

Hoekstra, A.

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

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 Stat. Nonlin. Soft Matter Phys. 82(3 Pt 2), 036703 (2010).
[Crossref] [PubMed]

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 558–589 (2007).
[Crossref]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 546–557 (2007).
[Crossref]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A 23(10), 2592–2601 (2006).
[Crossref] [PubMed]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A 23(10), 2578–2591 (2006).
[Crossref] [PubMed]

Iatì, M. A.

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. I. Sindoni, “Optical resonances of spheres containing an eccentric spherical inclusion Résonances optiques de sphères contenant une inclusion sphérique excentrée,” J. Opt. 29(1), 28–34 (1998).
[Crossref]

Johnson, B. R.

Kiehl, J. T.

P. Chýlek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18(5), 2229–2233 (1978).
[Crossref]

Kliewer, K. L.

Ko, M. K. W.

P. Chýlek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18(5), 2229–2233 (1978).
[Crossref]

Krieger, U. K.

A. Moridnejad, T. C. Preston, and U. K. Krieger, “Tracking water sorption in glassy aerosol particles using morphology-dependent resonances,” J. Phys. Chem. A 121(42), 8176–8184 (2017).
[Crossref] [PubMed]

Lacis, A. A.

M. I. Mishchenko and A. A. Lacis, “Morphology-dependent resonances of nearly spherical particles in random orientation,” Appl. Opt. 42(27), 5551–5556 (2003).
[Crossref] [PubMed]

M. I. Mishchenko and A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116(1–2), 167–179 (2000).
[Crossref]

Lam, C. C.

Leung, P. T.

Li, J.

Liu, C.

C. Liu, S. Teng, Y. Zhu, M. A. Yurkin, and Y. L. Yung, “Performance of the discrete dipole approximation for optical properties of black carbon aggregates,” J. Quant. Spectrosc. Radiat. Transf. 221, 98–109 (2018).
[Crossref]

C. Liu, L. Bi, R. L. Panetta, P. Yang, and M. A. Yurkin, “Comparison between the pseudo-spectral time domain method and the discrete dipole approximation for light scattering simulations,” Opt. Express 20(15), 16763–16776 (2012).
[Crossref]

Lumme, K.

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

Maltsev, V. P.

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 Stat. Nonlin. Soft Matter Phys. 82(3 Pt 2), 036703 (2010).
[Crossref] [PubMed]

Mishchenko, M. I.

M. I. Mishchenko and A. A. Lacis, “Morphology-dependent resonances of nearly spherical particles in random orientation,” Appl. Opt. 42(27), 5551–5556 (2003).
[Crossref] [PubMed]

M. I. Mishchenko and A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116(1–2), 167–179 (2000).
[Crossref]

Moridnejad, A.

A. Moridnejad, T. C. Preston, and U. K. Krieger, “Tracking water sorption in glassy aerosol particles using morphology-dependent resonances,” J. Phys. Chem. A 121(42), 8176–8184 (2017).
[Crossref] [PubMed]

Muinonen, K.

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

Ngo, D.

Panetta, R. L.

Penttilä, A.

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

Pinnick, R. G.

Preston, T. C.

A. Moridnejad, T. C. Preston, and U. K. Krieger, “Tracking water sorption in glassy aerosol particles using morphology-dependent resonances,” J. Phys. Chem. A 121(42), 8176–8184 (2017).
[Crossref] [PubMed]

Rahola, J.

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

A. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. 37(36), 8482–8497 (1998).
[Crossref] [PubMed]

Rushforth, C. K.

Saija, R.

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. I. Sindoni, “Optical resonances of spheres containing an eccentric spherical inclusion Résonances optiques de sphères contenant une inclusion sphérique excentrée,” J. Opt. 29(1), 28–34 (1998).
[Crossref]

Samokhin, A. B.

N. V. Budko and A. B. Samokhin, “Spectrum of the volume integral operator of electromagnetic scattering,” SIAM J. Sci. Comput. 28(2), 682–700 (2006).
[Crossref]

Shkuratov, Y.

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

Sindoni, O. I.

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. I. Sindoni, “Optical resonances of spheres containing an eccentric spherical inclusion Résonances optiques de sphères contenant une inclusion sphérique excentrée,” J. Opt. 29(1), 28–34 (1998).
[Crossref]

Sloot, P.

Smunev, D. A.

D. A. Smunev, P. C. Chaumet, and M. A. Yurkin, “Rectangular dipoles in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transf. 156, 67–79 (2015).
[Crossref]

Teng, S.

C. Liu, S. Teng, Y. Zhu, M. A. Yurkin, and Y. L. Yung, “Performance of the discrete dipole approximation for optical properties of black carbon aggregates,” J. Quant. Spectrosc. Radiat. Transf. 221, 98–109 (2018).
[Crossref]

Videen, G.

Yang, P.

Young, K.

Yung, Y. L.

C. Liu, S. Teng, Y. Zhu, M. A. Yurkin, and Y. L. Yung, “Performance of the discrete dipole approximation for optical properties of black carbon aggregates,” J. Quant. Spectrosc. Radiat. Transf. 221, 98–109 (2018).
[Crossref]

Yurkin, M. A.

C. Liu, S. Teng, Y. Zhu, M. A. Yurkin, and Y. L. Yung, “Performance of the discrete dipole approximation for optical properties of black carbon aggregates,” J. Quant. Spectrosc. Radiat. Transf. 221, 98–109 (2018).
[Crossref]

D. A. Smunev, P. C. Chaumet, and M. A. Yurkin, “Rectangular dipoles in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transf. 156, 67–79 (2015).
[Crossref]

M. A. Yurkin, “Symmetry relations for the Mueller scattering matrix integrated over the azimuthal angle,” J. Quant. Spectrosc. Radiat. Transf. 131, 82–87 (2013).
[Crossref]

C. Liu, L. Bi, R. L. Panetta, P. Yang, and M. A. Yurkin, “Comparison between the pseudo-spectral time domain method and the discrete dipole approximation for light scattering simulations,” Opt. Express 20(15), 16763–16776 (2012).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

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 Stat. Nonlin. Soft Matter Phys. 82(3 Pt 2), 036703 (2010).
[Crossref] [PubMed]

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 546–557 (2007).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 558–589 (2007).
[Crossref]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A 23(10), 2592–2601 (2006).
[Crossref] [PubMed]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A 23(10), 2578–2591 (2006).
[Crossref] [PubMed]

Zhu, Y.

C. Liu, S. Teng, Y. Zhu, M. A. Yurkin, and Y. L. Yung, “Performance of the discrete dipole approximation for optical properties of black carbon aggregates,” J. Quant. Spectrosc. Radiat. Transf. 221, 98–109 (2018).
[Crossref]

Zubko, E.

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

Ann. Phys. (1)

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[Crossref]

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[Crossref]

Appl. Opt. (2)

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A 23(10), 2592–2601 (2006).
[Crossref] [PubMed]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. I. Theoretical analysis,” J. Opt. Soc. Am. A 23(10), 2578–2591 (2006).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. A (1)

A. Moridnejad, T. C. Preston, and U. K. Krieger, “Tracking water sorption in glassy aerosol particles using morphology-dependent resonances,” J. Phys. Chem. A 121(42), 8176–8184 (2017).
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J. Quant. Spectrosc. Radiat. Transf. (7)

A. Penttilä, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, “Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 417–436 (2007).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 558–589 (2007).
[Crossref]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 546–557 (2007).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

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[Crossref]

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[Crossref]

C. Liu, S. Teng, Y. Zhu, M. A. Yurkin, and Y. L. Yung, “Performance of the discrete dipole approximation for optical properties of black carbon aggregates,” J. Quant. Spectrosc. Radiat. Transf. 221, 98–109 (2018).
[Crossref]

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Figures (13)

Fig. 1
Fig. 1 Extinction efficiency (Qext) and asymmetry factor (g) of spheres with m = 1.6 given by the Mie simulations with different range of x.
Fig. 2
Fig. 2 Comparison of the scattering matrix elements of spheres with m = 1.6 and x around 20. The green lines characterize the phase matrix elements of the sphere with x = 20.0001, and the red and blue lines characterize the ones with x = 19.9981 and 20.0021, respectively.
Fig. 3
Fig. 3 Extinction efficiency calculated by the Mie theory versus x with m varying from 1.2 to 1.6 with a step of 0.2.
Fig. 4
Fig. 4 DDA simulations of the extinction efficiency versus x, corresponding to b 55 (1) MDR (with m = 1.2). Markers indicate simulated data (with n from 128 to 768) and the solid lines show the Lorentzian fits with a cubic baseline (see text). Mie results are also shown together with a fit; the corresponding two lines overlap on the whole range.
Fig. 5
Fig. 5 Convergence of the peak parameters (xc, w, and A) from the DDA simulations, shown in Fig. 4 (for the MDR with m = 1.2 and x = 50.5369), with refining discretization. Solid lines correspond to the fits of c0 + c2n−2 + c3n−3 in three different ranges of n, extrapolated to 1/n→ 0 (see text for details). The insets show the results for 6 largest grids.
Fig. 6
Fig. 6 Same as Fig. 5 but for the baseline parameters (y0, y1, y2, and y3).
Fig. 7
Fig. 7 Comparison of the P11 and P12 elements at the MDR with m = 1.2 and x = 50.5369. Shown are the Mie results and DDA simulations with n = 768 and 192.
Fig. 8
Fig. 8 DDA simulations of the extinction efficiency versus x, corresponding to b 17 (1) MDR with m = 1.6. Markers indicate simulated data (three representative n from 128 to 1024) and the solid lines show the fits by two Lorentzians with a cubic baseline (see text). Mie results are also shown together with a single-peak fit (solid line for simulated data and dashed line for fit); the corresponding two lines overlap on the whole range.
Fig. 9
Fig. 9 Same as Fig. 8, but showing the results for all levels of discretization and in the reduced range of x.
Fig. 10
Fig. 10 Comparison of the original parameters of two peaks derived from the DDA simulations, shown in Figs. 8 and 9 (for the MDR with m = 1.6 and x = 13.24417), to that of the averaged peak (see text for details). Parts (a), (b), and (c) show the peak positions, widths, and integrals, respectively.
Fig. 11
Fig. 11 Convergence of the average peak parameters (x0, w0, and A0) derived from the DDA simulations for the MDR with m = 1.6 and x = 13.24417, with refining discretization (same as in Fig. 10). Solid lines correspond to the fits of c0 + c1n−1 + c2n−2 in three different ranges of n, extrapolated to 1/n→ 0 (see text for details).
Fig. 12
Fig. 12 Same as Fig. 11 but for the baseline parameters (y0, y1, y2, and y3) and using other ranges of n.
Fig. 13
Fig. 13 Convergence of the average peak parameters (x0, w0, and A0) derived by two-peaks fit from the DDA simulations, shown in Fig. 4 (for the MDR with m = 1.2 and x = 50.5369), with refining discretization. Solid lines correspond to the fits of c0 + c2n−2 + c3n−3 in two different ranges of n, extrapolated to 1/n→ 0 (the same procedure as in Fig. 5).

Tables (4)

Tables Icon

Table 1 The absolute errors of the extrapolated values of both peak and baseline parameters for the MDR with m = 1.2 and x = 50.5369 (see Figs. 5 and 6) using three different grid ranges of DDA simulations in comparison with the Mie result. The values for n = 768 are from the direct fit of the DDA results.

Tables Icon

Table 2 The absolute errors of the extrapolated values of average peak parameters for the MDR with m = 1.6 and x = 13.24417 (see Fig. 11) using different grid ranges of DDA simulations in comparison with the Mie result. The values for n = 1024 are from the direct fit of the DDA results.

Tables Icon

Table 3 Same as Table 2 but for the baseline parameters (y0, y1, y2, and y3) and using other ranges of n (the data is in Fig. 12).

Tables Icon

Table 4 Same as Table 3, but for the MDR with m = 1.2 and x = 50.5369, based on the data in Fig. 13.

Equations (4)

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Q ext = 2 x 2 n=1 (2n+1)Re( a n + b n ) .
b n = m ψ n ' (mx) ψ n (x) ψ n (mx) ψ n ' (x) m ψ n ' (mx) ξ n (x) ψ n (mx) ξ n ' (x) ,
y(x)= y 0 + y 1 (x x c )+ y 2 (x x c ) 2 + y 3 (x x c ) 3 + 2A π w 4 (x x c ) 2 + w 2 ,
y(x)= y 0 + y 1 (x x 0 )+ y 2 (x x 0 ) 2 + y 3 (x x 0 ) 3 + 2 A 1 π w 1 4 (x x c1 ) 2 + w 1 2 + 2 A 2 π w 2 4 (x x c2 ) 2 + w 2 2 ,

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