Modeling optical properties of particles with small-scale surface roughness: combination of group theory with a perturbation approach

Michael Kahnert; Tom Rother

doi:10.1364/OE.19.011138

Modeling optical properties of particles with small-scale surface roughness: combination of group theory with a perturbation approach

Michael Kahnert and Tom Rother

Optics Express
Vol. 19,
Issue 12,
pp. 11138-11151
(2011)
•https://doi.org/10.1364/OE.19.011138

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Michael Kahnert and Tom Rother, "Modeling optical properties of particles with small-scale surface roughness: combination of group theory with a perturbation approach," Opt. Express 19, 11138-11151 (2011)

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Related Topics
Table of Contents Category
- Scattering
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Aerosols (010.1110)
- Atmospheric optics (010.1290)
- Aerosol and cloud effects (290.1090)
- Backscattering (290.1350)
- Scattering, particles (290.5850)
- Scattering, rough surfaces (290.5880)
- Scattering theory (290.5825)

About this Article
History
- Original Manuscript: April 27, 2011
- Revised Manuscript: May 17, 2011
- Manuscript Accepted: May 17, 2011
- Published: May 23, 2011
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 6, Iss. 7

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Open Access

Abstract

A T-matrix method for scattering by particles with small-scale surface roughness is presented. The method combines group theory with a perturbation expansion approach. Group theory is found to reduce CPU-time by 4–6 orders of magnitude. The perturbation expansion extends the range of size parameters by a factor of 5 compared to non-perturbative methods. An application to optically hard particles shows that small-scale surface roughness changes scattering in side- and backscattering directions, and it impacts the single-scattering albedo. This can have important implications for interpreting remote sensing observations, and for the climate impact of mineral aerosols.

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References

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|
Year
|
Author
|
Publication

M. Kahnert, T. Nousiainen, and P. Mauno, “On the impact of small-scale surface roughness and non-sphericity on the optical properties of hematite aerosols,” J. Quant. Spectrosc. Radiat. Transfer (to be published).
T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gaeyt, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
[Crossref] [PubMed]
J.-C. Auger, G. Fernandes, K. Aptowicz, Y.-L. Pan, and R. Chang, “Influence of surface roughness on the elastic-light scattering patterns of micron-sized aerosol particles,” Appl. Phys. B 99, 229–234 (2010).
[Crossref]
M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transfer 112, 671–692 (2011).
[Crossref]
K. Muinonen, “Light scattering by stochastically shaped particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic Press, 2000), pp. 323–354.
[Crossref]
F. M. Kahnert, J. J. Stamnes, and K. Stamnes, “Application of the extended boundary condition method to homogeneous particles with point group symmetries,” Appl. Opt. 40, 3110–3123 (2001).
[Crossref]
M. Kahnert, “Irreducible representations of finite groups in the T matrix formulation of the electromagnetic scattering problem,” J. Opt. Soc. Am. A 22, 1187–1199 (2005).
[Crossref]
M. Kahnert, “Boundary symmetries in linear differential and integral equation problems applied to the self-consistent Green’s function formalism of acoustic and electromagnetic scattering,” Opt. Commun. 265, 383–393 (2006).
[Crossref]
P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]
F. M. Schulz, K. Stamnes, and J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T-matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[Crossref]
T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1019–1029 (2003).
[Crossref]
T. Rother and J. Wauer, “Case study about the accuracy behavior of three different T-matrix methods,” Appl. Opt. 49, 5746–5756 (2010).
[Crossref] [PubMed]
W. Greiner and J. Reinhardt, Quantum Electrodynamics (Springer, 2008).
T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).
[Crossref]
F. M. Kahnert, J. J. Stamnes, and K. Stamnes, “Application of the extended boundary condition method to particles with sharp edges: a comparison of two different surface integration approaches,” Appl. Opt. 40, 3101–3109 (2001).
[Crossref]
O. Muñoz, H. Volten, J. W. Hovenier, M. Min, Y. G. Shkuratov, J. P. Jalava, W. J. van der Zande, and L. B. F. M. Waters, “Experimental and computational study of light scattering by irregular particles with extreme refractive indices: hematite and rutile,” Astron. Astrophys. 446, 525–535 (2006).
[Crossref]
D. Petrov, Y. Shkuvatov, and G. Videen, “Analytical light-scattering solution for Chebyshev particles,” J. Opt. Soc. Am. A 24, 1103–1119 (2007).
[Crossref]

2011 (1)

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transfer 112, 671–692 (2011).
[Crossref]

2010 (2)

J.-C. Auger, G. Fernandes, K. Aptowicz, Y.-L. Pan, and R. Chang, “Influence of surface roughness on the elastic-light scattering patterns of micron-sized aerosol particles,” Appl. Phys. B 99, 229–234 (2010).
[Crossref]

T. Rother and J. Wauer, “Case study about the accuracy behavior of three different T-matrix methods,” Appl. Opt. 49, 5746–5756 (2010).
[Crossref] [PubMed]

2007 (1)

D. Petrov, Y. Shkuvatov, and G. Videen, “Analytical light-scattering solution for Chebyshev particles,” J. Opt. Soc. Am. A 24, 1103–1119 (2007).
[Crossref]

2006 (3)

O. Muñoz, H. Volten, J. W. Hovenier, M. Min, Y. G. Shkuratov, J. P. Jalava, W. J. van der Zande, and L. B. F. M. Waters, “Experimental and computational study of light scattering by irregular particles with extreme refractive indices: hematite and rutile,” Astron. Astrophys. 446, 525–535 (2006).
[Crossref]

T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gaeyt, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
[Crossref] [PubMed]

M. Kahnert, “Boundary symmetries in linear differential and integral equation problems applied to the self-consistent Green’s function formalism of acoustic and electromagnetic scattering,” Opt. Commun. 265, 383–393 (2006).
[Crossref]

2005 (1)

M. Kahnert, “Irreducible representations of finite groups in the T matrix formulation of the electromagnetic scattering problem,” J. Opt. Soc. Am. A 22, 1187–1199 (2005).
[Crossref]

2003 (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1019–1029 (2003).
[Crossref]

2001 (2)

F. M. Kahnert, J. J. Stamnes, and K. Stamnes, “Application of the extended boundary condition method to particles with sharp edges: a comparison of two different surface integration approaches,” Appl. Opt. 40, 3101–3109 (2001).
[Crossref]

F. M. Kahnert, J. J. Stamnes, and K. Stamnes, “Application of the extended boundary condition method to homogeneous particles with point group symmetries,” Appl. Opt. 40, 3110–3123 (2001).
[Crossref]

1998 (1)

F. M. Schulz, K. Stamnes, and J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T-matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[Crossref]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

Aptowicz, K.

Auger, J.-C.

Cairns, B.

Chang, R.

Dlugach, J. M.

Fernandes, G.

Gaeyt, J.-F.

T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gaeyt, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
[Crossref] [PubMed]

Greiner, W.

W. Greiner and J. Reinhardt, Quantum Electrodynamics (Springer, 2008).

Heckenberg, N. R.

Hovenier, J. W.

Jalava, J. P.

Kahnert, F. M.

Kahnert, M.

M. Kahnert, “Irreducible representations of finite groups in the T matrix formulation of the electromagnetic scattering problem,” J. Opt. Soc. Am. A 22, 1187–1199 (2005).
[Crossref]

M. Kahnert, T. Nousiainen, and P. Mauno, “On the impact of small-scale surface roughness and non-sphericity on the optical properties of hematite aerosols,” J. Quant. Spectrosc. Radiat. Transfer (to be published).

Kiselev, N. N.

Liu, L.

Mauno, P.

Min, M.

Mishchenko, M. I.

Muinonen, K.

K. Muinonen, “Light scattering by stochastically shaped particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic Press, 2000), pp. 323–354.
[Crossref]

Muñoz, O.

Nieminen, T. A.

Nousiainen, T.

Pan, Y.-L.

Petrov, D.

D. Petrov, Y. Shkuvatov, and G. Videen, “Analytical light-scattering solution for Chebyshev particles,” J. Opt. Soc. Am. A 24, 1103–1119 (2007).
[Crossref]

Reinhardt, J.

W. Greiner and J. Reinhardt, Quantum Electrodynamics (Springer, 2008).

Rosenbush, V. K.

Rother, T.

T. Rother and J. Wauer, “Case study about the accuracy behavior of three different T-matrix methods,” Appl. Opt. 49, 5746–5756 (2010).
[Crossref] [PubMed]

T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gaeyt, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
[Crossref] [PubMed]

T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).
[Crossref]

Rubinsztein-Dunlop, H.

Schmidt, K.

T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gaeyt, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
[Crossref] [PubMed]

Schulz, F. M.

Shcherbakov, V.

T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gaeyt, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
[Crossref] [PubMed]

Shkuratov, Y. G.

Shkuvatov, Y.

D. Petrov, Y. Shkuvatov, and G. Videen, “Analytical light-scattering solution for Chebyshev particles,” J. Opt. Soc. Am. A 24, 1103–1119 (2007).
[Crossref]

Stamnes, J. J.

Stamnes, K.

Tishkovets, V. P.

Travis, L. D.

van der Zande, W. J.

Videen, G.

D. Petrov, Y. Shkuvatov, and G. Videen, “Analytical light-scattering solution for Chebyshev particles,” J. Opt. Soc. Am. A 24, 1103–1119 (2007).
[Crossref]

Volten, H.

Waterman, P. C.

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

Waters, L. B. F. M.

Wauer, J.

T. Rother and J. Wauer, “Case study about the accuracy behavior of three different T-matrix methods,” Appl. Opt. 49, 5746–5756 (2010).
[Crossref] [PubMed]

T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gaeyt, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
[Crossref] [PubMed]

Appl. Opt. (5)

T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gaeyt, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
[Crossref] [PubMed]

T. Rother and J. Wauer, “Case study about the accuracy behavior of three different T-matrix methods,” Appl. Opt. 49, 5746–5756 (2010).
[Crossref] [PubMed]

Appl. Phys. B (1)

Astron. Astrophys. (1)

J. Opt. Soc. Am. A (2)

D. Petrov, Y. Shkuvatov, and G. Videen, “Analytical light-scattering solution for Chebyshev particles,” J. Opt. Soc. Am. A 24, 1103–1119 (2007).
[Crossref]

M. Kahnert, “Irreducible representations of finite groups in the T matrix formulation of the electromagnetic scattering problem,” J. Opt. Soc. Am. A 22, 1187–1199 (2005).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (2)

Opt. Commun. (1)

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

Other (4)

W. Greiner and J. Reinhardt, Quantum Electrodynamics (Springer, 2008).

T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles (Springer, 2009).
[Crossref]

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Fig. 1 2D (left) and 3D (right) Chebyshev particle with polynomial order ℓ=45 and deformation parameter ε =0.05.

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Fig. 2 Top row: hh (left) and vv (right) components of the polarized differential scattering cross sections of 2D Chebyshev particles in a fixed orientation, computed with the group theoretical/perturbative T-matrix approach (black) and with mieschka (red). Bottom row: hh (left) and vv (right) components computed with mieschka for two different orientations.

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Fig. 3 Optical properties of spheres (dashed line), and 3D Chebyshev particles with ε =0.01 (blue) and A=0.11λ (red): ω (top left), g (top right), and C _bak (bottom left). Also shown is the CPU time (bottom right) for computations using group theory (circles) and not using group theory (pluses).

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Fig. 4 Mueller matrix elements F ₁₁ (left column) and –F ₁₂ /F ₁₁ (right column) of 3D Chebyshev particles (red) and spheres (blue) with particle radius 1 μm (top row) and 6 μm (bottom row).

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Tables (3)

Table 1 Reciprocity Test for 2D Chebyshev Particles

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Table 2 Reciprocity Test for 3D Chebyshev Particles with ε = 0.01

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Table 3 Reciprocity Test for 3D Chebyshev Particles with A = 0.11λ

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Equations (16)

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

r (θ) = r_{0} [1 + ɛ cos (ℓ θ)],

Λ = \frac{2 π r_{0}}{ℓ}

A = ɛ r_{0} .

r (θ, ϕ) = r_{0} [1 + ɛ cos (ℓ θ) cos (ℓ ϕ)],

T = U (g) \cdot T \cdot U^{- 1} (g),

[T, U (g)] = 0, \forall g \in 𝒢

T_{n, m, τ, n^{'}, m^{'}, τ^{'}} = 0 unless | m - m' | = 0, ℓ, 2 ℓ, \dots

T_{n, m, τ, n^{'}, m^{'}, τ^{'}} = {(- 1)}^{n + n'} T_{n, - m, τ, n', - m', τ'}

T_{n, m, τ, n^{'}, m^{'}, τ^{'}} = 0 unless (n + m + τ + n' + m' + τ') . even

T = - Rg Q \cdot Q^{- 1} .

T \cdot (Q_{0} + Δ Q) = - R g Q .

T = - (Rg Q + T \cdot Δ Q) \cdot Q_{0}^{- 1} .

T^{(0)} = - Rg Q \cdot Q_{0}^{- 1} .

T^{(n)} = - (Rg Q + T^{(n - 1)} \cdot Δ Q) \cdot Q_{0}^{- 1} .

{(\frac{d σ}{d Ω})}_{α, β} ({\hat{k}}_{inc}, {\hat{k}}_{sca}) = {(\frac{d σ}{d Ω})}_{β, α} (- {\hat{k}}_{sca}, - {\hat{k}}_{inc}) .

{(\frac{d σ}{d Ω})}_{α, β} (Θ = 90 °; θ_{p} = 0) = {(\frac{d σ}{d Ω})}_{β, α} (Θ = 270 °; θ_{p} = 90 °) .

r ₀	S_h,h (90°;0°)	S_h,h (270°;90°)	S_v,v (90°;0°)	S_v,v (270°;90°)
1	3.20368	3.20363	10.6317	10.6317
2	13.6312	13.6318	33.6934	33.6934
3	24.6851	24.6859	68.8825	68.8823
4	38.1723	38.1669	105.215	105.215
5	50.1139	50.1217	133.364	133.364
6	55.0429	55.0444	146.434	146.433
7	51.8876	51.8692	140.824	140.826

r ₀	S_h,h (90°;0°)	S_h,h (270°;90°)	S_v,v (90°;0°)	S_v,v (270°;90°)
1	1.41650	1.41372	4.39453	4.39484
2	5.00893	5.01386	11.0705	11.0697
3	8.70649	8.71068	25.8196	25.8173
4	16.3797	16.3665	46.2344	46.2356
5	27.4014	27.4142	72.0140	72.0139
6	39.1350	39.1371	103.497	103.497
7	51.8876	51.8692	140.824	140.826

r ₀	S_h,h (90°;0°)	S_h,h (270°;90°)	S_v,v (90°;0°)	S_v,v (270°;90°)
1	3.20368	3.20363	10.6317	10.6317
2	13.6312	13.6318	33.6934	33.6934
3	24.6851	24.6859	68.8825	68.8823
4	38.1723	38.1669	105.215	105.215
5	50.1139	50.1217	133.364	133.364
6	55.0429	55.0444	146.434	146.433
7	51.8876	51.8692	140.824	140.826

r ₀	S_h,h (90°;0°)	S_h,h (270°;90°)	S_v,v (90°;0°)	S_v,v (270°;90°)
1	1.41650	1.41372	4.39453	4.39484
2	5.00893	5.01386	11.0705	11.0697
3	8.70649	8.71068	25.8196	25.8173
4	16.3797	16.3665	46.2344	46.2356
5	27.4014	27.4142	72.0140	72.0139
6	39.1350	39.1371	103.497	103.497
7	51.8876	51.8692	140.824	140.826

Abstract

References

2011 (1)

2010 (2)

2007 (1)

2006 (3)

2005 (1)

2003 (1)

2001 (2)

1998 (1)

1965 (1)

Aptowicz, K.

Auger, J.-C.

Cairns, B.

Chang, R.

Dlugach, J. M.

Fernandes, G.

Gaeyt, J.-F.

Greiner, W.

Heckenberg, N. R.

Hovenier, J. W.

Jalava, J. P.

Kahnert, F. M.

Kahnert, M.

Kiselev, N. N.

Liu, L.

Mauno, P.

Min, M.

Mishchenko, M. I.

Muinonen, K.

Muñoz, O.

Nieminen, T. A.

Nousiainen, T.

Pan, Y.-L.

Petrov, D.

Reinhardt, J.

Rosenbush, V. K.

Rother, T.

Rubinsztein-Dunlop, H.

Schmidt, K.

Schulz, F. M.

Shcherbakov, V.

Shkuratov, Y. G.

Shkuvatov, Y.

Stamnes, J. J.

Stamnes, K.

Tishkovets, V. P.

Travis, L. D.

van der Zande, W. J.

Videen, G.

Volten, H.

Waterman, P. C.

Waters, L. B. F. M.

Wauer, J.

Appl. Opt. (5)

Appl. Phys. B (1)

Astron. Astrophys. (1)

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transfer (2)

Opt. Commun. (1)

Proc. IEEE (1)

Other (4)

Cited By

Figures (4)

Tables (3)

Equations (16)

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