Abstract

A new physical-geometric optics method is developed to compute the single-scattering properties of faceted particles. It incorporates a general absorption vector to accurately account for inhomogeneous wave effects, and subsequently yields the relevant analytical formulas effective and computationally efficient for absorptive scattering particles. A bundle of rays incident on a certain facet can be traced as a single beam. For a beam incident on multiple facets, a systematic beam-splitting technique based on computer graphics is used to split the original beam into several sub-beams so that each sub-beam is incident only on an individual facet. The new beam-splitting technique significantly reduces the computational burden. The present physical-geometric optics method can be generalized to arbitrary faceted particles with either convex or concave shapes and with a homogeneous or an inhomogeneous (e.g., a particle with a core) composition. The single-scattering properties of irregular convex homogeneous and inhomogeneous hexahedra are simulated and compared to their counterparts from two other methods including a numerically rigorous method.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2017 (3)

M. I. Mishchenko and M. A. Yurkin, “On the concept of random orientation in far-field electromagnetic scattering by nonspherical particles,” Opt. Lett. 42(3), 494–497 (2017).
[Crossref] [PubMed]

L. Bi and P. Yang, “Improved ice particle optical property simulations in the ultraviolet to far-infrared regime,” J. Quant. Spectrosc. Radiat. Transf. 189, 228–237 (2017).
[Crossref]

G. Xu, B. Sun, S. D. Brooks, P. Yang, G. W. Kattawar, and X. Zhang, “Modeling the inherent optical properties of aquatic particles using an irregular hexahedral ensemble,” J. Quant. Spectrosc. Radiat. Transf. 191, 30–39 (2017).
[Crossref]

2016 (3)

B. Sun, G. W. Kattawar, P. Yang, and K. F. Ren, “Rigorous 3-D vectorial complex ray model applied to light scattering by an arbitrary spheroid,” J. Quant. Spectrosc. Radiat. Transf. 179, 1–10 (2016).
[Crossref]

B. Sun, G. W. Kattawar, P. Yang, M. S. Twardowski, and J. M. Sullivan, “Simulation of the scattering properties of a chain-forming triangular prism oceanic diatom,” J. Quant. Spectrosc. Radiat. Transf. 178, 390–399 (2016).
[Crossref]

G. Tang, P. Yang, B. Sun, R. L. Panetta, and G. W. Kattawar, “Enhancement of the computational efficiency of the near-to-far field mapping in the finite-difference method and ray-by-ray method with the fast multi-pole plane wave expansion approach,” J. Quant. Spectrosc. Radiat. Transf. 176, 70–81 (2016).
[Crossref]

2015 (1)

J. Liu, P. Yang, and K. Muinonen, “Dust-aerosol optical modeling with Gaussian spheres: Combined invariant-imbedding T-matrix and geometric-optics approach,” J. Quant. Spectrosc. Radiat. Transf. 161, 136–144 (2015).
[Crossref]

2014 (2)

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

2013 (1)

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

2011 (1)

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

2007 (1)

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]

2005 (1)

P. C. Chang, J. Walker, and K. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3-4), 327–341 (2005).
[Crossref]

1998 (2)

M. I. Mishchenko and L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45(4), 1044–1055 (1998).
[Crossref] [PubMed]

1997 (1)

1996 (3)

1995 (2)

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34(21), 4573–4588 (1995).
[Crossref] [PubMed]

J. Song and W. C. Chew, “Multilevel fast‐multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10(1), 14–19 (1995).
[Crossref]

1994 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

1991 (1)

1988 (3)

1982 (1)

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

1971 (1)

P. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields 3(4), 825–839 (1971).
[Crossref]

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagation 14(3), 302–307 (1966).
[Crossref]

1965 (1)

P. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20(1), 33–40 (1962).
[Crossref]

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15–24 (1961).
[Crossref]

1954 (1)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12(1), 13–23 (1954).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Baum, B. A.

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

Bi, L.

L. Bi and P. Yang, “Improved ice particle optical property simulations in the ultraviolet to far-infrared regime,” J. Quant. Spectrosc. Radiat. Transf. 189, 228–237 (2017).
[Crossref]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

Borovoi, A.

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

Brooks, S. D.

G. Xu, B. Sun, S. D. Brooks, P. Yang, G. W. Kattawar, and X. Zhang, “Modeling the inherent optical properties of aquatic particles using an irregular hexahedral ensemble,” J. Quant. Spectrosc. Radiat. Transf. 191, 30–39 (2017).
[Crossref]

Cai, Q.

Chang, P. C.

P. C. Chang, J. Walker, and K. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3-4), 327–341 (2005).
[Crossref]

Chew, W. C.

J. Song and W. C. Chew, “Multilevel fast‐multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10(1), 14–19 (1995).
[Crossref]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20(1), 33–40 (1962).
[Crossref]

Draine, B. T.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

Friedman, B.

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12(1), 13–23 (1954).
[Crossref]

Fuller, K. A.

Hoekstra, A. G.

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]

Hopcraft, K.

P. C. Chang, J. Walker, and K. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3-4), 327–341 (2005).
[Crossref]

Hu, Y.

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

Johnson, B. R.

Kattawar, G. W.

G. Xu, B. Sun, S. D. Brooks, P. Yang, G. W. Kattawar, and X. Zhang, “Modeling the inherent optical properties of aquatic particles using an irregular hexahedral ensemble,” J. Quant. Spectrosc. Radiat. Transf. 191, 30–39 (2017).
[Crossref]

B. Sun, G. W. Kattawar, P. Yang, and K. F. Ren, “Rigorous 3-D vectorial complex ray model applied to light scattering by an arbitrary spheroid,” J. Quant. Spectrosc. Radiat. Transf. 179, 1–10 (2016).
[Crossref]

G. Tang, P. Yang, B. Sun, R. L. Panetta, and G. W. Kattawar, “Enhancement of the computational efficiency of the near-to-far field mapping in the finite-difference method and ray-by-ray method with the fast multi-pole plane wave expansion approach,” J. Quant. Spectrosc. Radiat. Transf. 176, 70–81 (2016).
[Crossref]

B. Sun, G. W. Kattawar, P. Yang, M. S. Twardowski, and J. M. Sullivan, “Simulation of the scattering properties of a chain-forming triangular prism oceanic diatom,” J. Quant. Spectrosc. Radiat. Transf. 178, 390–399 (2016).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

K. A. Fuller and G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13(2), 90–92 (1988).
[Crossref] [PubMed]

Konoshonkin, A.

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

Kustova, N.

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

Liou, K.

Liou, K. N.

Liu, J.

J. Liu, P. Yang, and K. Muinonen, “Dust-aerosol optical modeling with Gaussian spheres: Combined invariant-imbedding T-matrix and geometric-optics approach,” J. Quant. Spectrosc. Radiat. Transf. 161, 136–144 (2015).
[Crossref]

Liu, Q. H.

Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45(4), 1044–1055 (1998).
[Crossref] [PubMed]

Macke, A.

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystals,” J Atmo. Sci. 53, 2813–2825 (1996).

Mackowski, D. W.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Mishchenko, M.

Mishchenko, M. I.

M. I. Mishchenko and M. A. Yurkin, “On the concept of random orientation in far-field electromagnetic scattering by nonspherical particles,” Opt. Lett. 42(3), 494–497 (2017).
[Crossref] [PubMed]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13(11), 2266–2278 (1996).
[Crossref]

Mueller, J.

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystals,” J Atmo. Sci. 53, 2813–2825 (1996).

Muinonen, K.

J. Liu, P. Yang, and K. Muinonen, “Dust-aerosol optical modeling with Gaussian spheres: Combined invariant-imbedding T-matrix and geometric-optics approach,” J. Quant. Spectrosc. Radiat. Transf. 161, 136–144 (2015).
[Crossref]

Panetta, R. L.

G. Tang, P. Yang, B. Sun, R. L. Panetta, and G. W. Kattawar, “Enhancement of the computational efficiency of the near-to-far field mapping in the finite-difference method and ray-by-ray method with the fast multi-pole plane wave expansion approach,” J. Quant. Spectrosc. Radiat. Transf. 176, 70–81 (2016).
[Crossref]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Raschke, E.

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystals,” J Atmo. Sci. 53, 2813–2825 (1996).

Ren, K. F.

B. Sun, G. W. Kattawar, P. Yang, and K. F. Ren, “Rigorous 3-D vectorial complex ray model applied to light scattering by an arbitrary spheroid,” J. Quant. Spectrosc. Radiat. Transf. 179, 1–10 (2016).
[Crossref]

Russek, J.

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12(1), 13–23 (1954).
[Crossref]

Song, J.

J. Song and W. C. Chew, “Multilevel fast‐multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10(1), 14–19 (1995).
[Crossref]

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15–24 (1961).
[Crossref]

Sullivan, J. M.

B. Sun, G. W. Kattawar, P. Yang, M. S. Twardowski, and J. M. Sullivan, “Simulation of the scattering properties of a chain-forming triangular prism oceanic diatom,” J. Quant. Spectrosc. Radiat. Transf. 178, 390–399 (2016).
[Crossref]

Sun, B.

G. Xu, B. Sun, S. D. Brooks, P. Yang, G. W. Kattawar, and X. Zhang, “Modeling the inherent optical properties of aquatic particles using an irregular hexahedral ensemble,” J. Quant. Spectrosc. Radiat. Transf. 191, 30–39 (2017).
[Crossref]

B. Sun, G. W. Kattawar, P. Yang, and K. F. Ren, “Rigorous 3-D vectorial complex ray model applied to light scattering by an arbitrary spheroid,” J. Quant. Spectrosc. Radiat. Transf. 179, 1–10 (2016).
[Crossref]

G. Tang, P. Yang, B. Sun, R. L. Panetta, and G. W. Kattawar, “Enhancement of the computational efficiency of the near-to-far field mapping in the finite-difference method and ray-by-ray method with the fast multi-pole plane wave expansion approach,” J. Quant. Spectrosc. Radiat. Transf. 176, 70–81 (2016).
[Crossref]

B. Sun, G. W. Kattawar, P. Yang, M. S. Twardowski, and J. M. Sullivan, “Simulation of the scattering properties of a chain-forming triangular prism oceanic diatom,” J. Quant. Spectrosc. Radiat. Transf. 178, 390–399 (2016).
[Crossref]

Tang, G.

G. Tang, P. Yang, B. Sun, R. L. Panetta, and G. W. Kattawar, “Enhancement of the computational efficiency of the near-to-far field mapping in the finite-difference method and ray-by-ray method with the fast multi-pole plane wave expansion approach,” J. Quant. Spectrosc. Radiat. Transf. 176, 70–81 (2016).
[Crossref]

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[Crossref]

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G. Xu, B. Sun, S. D. Brooks, P. Yang, G. W. Kattawar, and X. Zhang, “Modeling the inherent optical properties of aquatic particles using an irregular hexahedral ensemble,” J. Quant. Spectrosc. Radiat. Transf. 191, 30–39 (2017).
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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

G. Tang, P. Yang, B. Sun, R. L. Panetta, and G. W. Kattawar, “Enhancement of the computational efficiency of the near-to-far field mapping in the finite-difference method and ray-by-ray method with the fast multi-pole plane wave expansion approach,” J. Quant. Spectrosc. Radiat. Transf. 176, 70–81 (2016).
[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Diagrams showing the reflection-refraction events at order p = 0 and p>0. For p>0, the superscripts ‘s’, ‘r’, and ‘t’ represent the scattered, reflected and refracted quantities. β × α = e Here, the subscript ‘i’ represents the incident quantities. All other subscripts correspond to the reflection-refraction order p.
Fig. 2
Fig. 2 A systematic beam prism obtained from the p-th order to the (p + 1)-th order.
Fig. 3
Fig. 3 Translation, rotation, and projection process.
Fig. 4
Fig. 4 The schematic Weiler-Atherton algorithm.
Fig. 5
Fig. 5 Three hexahedra used in the comparisons. The panel (a) is marked as hexahedron 1 and the panel (b) is hexahedron 2. The normalized vertices are given in Table 2. The panel (c) is an inhomogeneous hexahedron with hexahedron 1 included with hexahedron 2.
Fig. 6
Fig. 6 Comparisons of the phase matrix elements of the hexahedron 2 in random orientation calculated by the PGOH and the present PGOM. The incident wavelength is 0.658µm and the equivalent-volume radius is 8.0µm. The refractive index is 1.12 + i0.0005.
Fig. 7
Fig. 7 Comparisons of the phase matrix elements of inhomogeneous hexahedra, calculated by the PGOH and the PGOM under the random orientation condition. The incident wavelength is 0.658 µm. The inner layer is hexahedron 1 and the equivalent-volume radius is 4.68 µm while the outer layer is hexahedron 2 and the corresponding radius is 8.0 µm. The refractive indices of the hexahedra 1 and 2 are 1.12 + i0.0005 and 1.0 for panel (a) and 1.12 + i0.0005 and 1.12 + i0.0005 for panel (b).
Fig. 8
Fig. 8 Comparisons of the phase matrix elements of hexahedron 2, calculated by the PGOM, the CGOM and the IITM under random orientation condition. The equivalent-volume radius is 8.0µm and the incident wavelength is 0.658µm. The refractive index is 1.12 + i0.0005.
Fig. 9
Fig. 9 Comparisons of the phase matrix elements of a hexagonal column with a unit aspect ratio (defined as the height over the diameter of the circumscribed circle of the bottom hexagon), calculated by the PGOM and the IITM under the random orientation condition. The height of the column is 20.94µm and the incident wavelength is 0.658µm. The refractive index is 1.33.
Fig. 10
Fig. 10 Comparisons of the phase matrix elements of the inhomogeneous hexahedron, calculated by the PGOM and the IITM under the random orientation condition. The incident wavelength is 0.658µm. The equivalent-volume radii of hexahedron 1 and 2 are 4.6 µm and 8.0 µm. The refractive indices of the inner layer and outer layer are 1.12 + i0.0005 and 1.03 + i0.005, respectively.

Tables (2)

Tables Icon

Table 1 Symbols used in the formulas in this paper.

Tables Icon

Table 2 Vertices of two hexahedra. The two shapes are normalized so that the radii of their circumscribed spheres are unity.

Equations (35)

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E s ( r )| r = exp( i k 0 r ) i k 0 r i k 0 3 4π V ( 1 m 2 )[ E ( r )( r ^ E ( r ) ) ]exp( i k 0 r ^ r ) d 3 r ,
[ E p,α ( r p,1 ) E p,β ( r p,1 ) ]= U p [ E α i E β i ]exp( i k 0 ϕ p,1 ),
ϕ p,1 = φ p,1 +i ψ p,1 .
φ 0,1 = e ^ i r 0,1 , ψ 0,1 =0,
φ p,1 = φ p1,1 + N r,p | r p,1 r p1,1 |,
ψ p,1 = ψ p1,1 + N i,p | r p,1 r p1,1 |,
N p = N r,p +i N i,p .
k p = k 0 ( N r,p e ^ p + A p ),
A 0 =0, A p = A p1 +( N i,p e ^ p A p1 ) n ^ p1 e ^ p n ^ p1 .
[ E p,α E p,β ( r p,1 + ω p ) ( r p,1 + ω p ) ]= U p [ E α i E β i ]exp[ i( k 0 ϕ p,1 + k p ω p ) ],
[ E p,α E p,β ( r ) ( r ) ]= U p [ E α i E β i ]exp[ i( k 0 ϕ p,1 + k p ω p + k p+1 e ^ p+1 l ) ],
r = r p,1 + ω p +l e ^ p+1 .
F p = N r,p 2 ( | U p 11 | 2 + | U p 12 | 2 + | U p 21 | 2 + | U p 22 | 2 )| e ^ p+1 n ^ p | D ˜ p ,
D ˜ p =exp( 2 k 0 φ p,1 ) Λ p exp( 2 k 0 A p ω p ) d 2 ω p = 1 2 k 0 2 j=1 N ν [ k 0 ( r p,j+1 r p,j )( A p × n ^ p ) A p A p ( A p n ^ p ) 2 ] [ exp( 2 k 0 φ p,j+1 )exp( 2 k 0 φ p,j ) 2 k 0 ( φ p,j+1 φ p,j ) ].
T( x L , y L , z L )=[ 1 0 0 x L 0 1 0 y L 0 0 1 z L 0 0 0 1 ],
R=[ R 3 0 0 1 ],
M proj =[ 1 0 e x / e z 0 0 1 e y / e z 0 0 0 0 0 0 0 0 1 ],
[ E α s E β s ]= exp( i k 0 r ) k 0 r i k 3 4π p=0 V p ( 1 m p+1 2 ) K p [ E p,α ( r ) E p,β ( r ) ]exp( i k 0 r ^ r ) d 3 r ,
K p =[ α ^ s α ^ p α ^ s β ^ p β ^ s α ^ p β ^ s β ^ p ].
[ E α s E β s ]= exp( i k 0 r ) k 0 r p=0 ( 1 m p+1 2 ) q p ( r ^ ) K p U p [ E α i E β i ],
q p ( r ^ )= i k 0 3 4π exp( i k 0 ϕ p,1 ) V p exp[ i( k p ω p + k p+1 e ^ p+1 l ) ] exp(i k 0 r ^ r ) d 3 r = i k 0 3 4π exp( i k 0 Φ p,1 ) Λ p d 2 ω p { | e ^ p+1 n ^ p |exp( i κ p ω p ) 0 [ r p+1,1 r p,1 ]+ ω p+1 n ^ p e ^ p+1 n ^ p dlexp[ i( κ p+1 e ^ p+1 )l ] } ,
Φ p,j = ϕ p,j r ^ r p,j ,
κ p = k p k 0 r ^ ,
k 0 Φ p+1,j = k 0 Φ p,j + κ p+1 ( r p+1,j r p,j ),
k 0 Φ p,j+1 = k 0 Φ p,j + κ p ( r p,j+1 r p,j ).
ω p+1 = ω p + ω p+1 n ^ p e ^ p+1 n ^ p e ^ p+1 ,
A p+1 ω p+1 = A p ω p + N i,p+1 ω p+1 n ^ p e ^ p+1 n ^ p .
q p ( r ^ )= k 0 ( | e ^ p+1 n ^ p+1 | D p+1 | e ^ p+1 n ^ p | D p ) κ p+1 e ^ p+1 ,
D p = k 0 2 4π exp( i k 0 Φ p,1 ) Λ p exp( i κ p ω p ) d 2 ω p = k 0 2 4π j=1 N ν [ ( r p,j+1 r p,j )( i κ p × n ^ p ) κ p κ p ( κ p n ^ p ) 2 ] [ exp( i k 0 Φ p,j+1 )exp( i k 0 Φ p,j ) k 0 ( Φ p,j+1 Φ p,j ) ].
Λ exp( c ω ) d 2 ω = j=1 N ν [ exp( c ν j+1 )exp( c ν j ) c c ( c n ^ ) 2 ] [ ( n ^ × c )( ν j+1 ν j ) c ( ν j+1 ν j ) ].
[ E α s E β s ]= exp( i k 0 r ) i k 0 r p=0 ( 1 m p+1 2 )[ | e ^ p n ^ p+1 | D p+1 | e ^ p n ^ p | D p ] K p U p κ p e ^ p [ E α i E β i ] .
S= p=0 ( 1 m p+1 2 )[ | e ^ p+1 n ^ p+1 | D p+1 | e ^ p+1 n ^ p | D p ] K p U p Γ κ p+1 e ^ p+1 =| e ^ 1 n ^ 0 | D 1 ( m 1 2 1) K 1 U 1 Γ κ 1 e ^ 1 + p=1 D p [ ( 1 m p 2 ) K p1 U p1 κ p e ^ p | e ^ p n ^ p | ( 1 m p+1 2 ) K p U p κ p+1 e ^ p+1 | e ^ p+1 n ^ p | ]Γ .
Γ=[ β ^ i β ^ s α i β ^ s α i β ^ s β ^ i β ^ s ].
σ ext =Im[ k 0 | E i | 2 V ( m 2 1 )( E ( r ) E i,* ( r ) ) d 3 r ] = 2π k 2 Re[ S 11 ( e ^ i )+ S 22 ( e ^ i ) ],
σ abs = k 0 | E i | 2 V ε i ( E ( r ) E * ( r ) ) d 3 r = 1 2 p=0 N r,p | U p | 2 [ | e ^ p n ^ p | D ˜ p | e ^ p n ^ p+1 | D ˜ p+1 ] .

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