Abstract

An exact transition matrix was formulated for electromagnetic scattering by an orthorhombic dielectric–magnetic sphere whose permeability dyadic is a scalar multiple of its permittivity dyadic. Calculations were made for plane waves incident on the sphere. As the size parameter increases, the role of anisotropy evolves; multiple lobes appear in the plots of the differential scattering efficiency in any scattering plane; the total scattering, extinction, and forward-scattering efficiencies exhibit a prominent maximum each; and the absorption efficiency generally increases with weak undulations. Certain orientations of the sphere with respect to the directions of propagation and the electric field of the incident plane wave make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection. Impedance match to the ambient free space decreases backscattering efficiency significantly, although anisotropy prevents null backscattering.

© 2013 Optical Society of America

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References

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  1. O. F. Mossotti, “Recherches théoriques sur l’induction électro-statique, envisagée d’après idées de Faraday,” Supp. Biblio. Univer. Genève Arch. Sci. Phys. Natur. 16, 193–198 (1847).
  2. O. F. Mossotti, “Discussione analitica sull influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricita alla superficie di più corpi elettrici disseminati in esso,” Mem. Mat. Fis. Modena 24, 49–74 (1850).
  3. M. Faraday, “Experimental relations of gold (and other materials) to light,” Philos. Trans. R. Soc. London 147, 145–181 (1857).
    [CrossRef]
  4. A. Walther, “Optical applications of solid glass spheres,” Ph.D. thesis (Delft University of Technology, 1959).
  5. “Spherical glass solar energy generator by andre rawlemon,” August25, 2012, http://www.designboom.com/technology/spherical-glass-solar-energy-generator-by-rawlemon/ (accessed on June 3, 2013).
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  7. M. Kerker, ed., Selected Papers on Light Scattering, Part 1 (SPIE, 1988).
  8. L. V. Lorenz, “Lysvevægelsen i og uden for en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Forh. 6, 1–62 (1890).
  9. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Lpz. 25, 377–445 (1908).
  10. G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions—translation,” Sandia Laboratories, Albuquerque, New Mexico, 1978, SAND78-6018. National Translation Center, Chicago, Illinois, Translation 79–21946.
  11. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, 1941), Chap. 7.
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.
  13. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
    [CrossRef]
  14. A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012).
    [CrossRef]
  15. T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).
  16. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991).
    [CrossRef]
  17. C. D. Gribble and A. J. Hall, Optical Mineralogy, Principles and Practice (University College London Press, 1992).
  18. J. C. Bose, “On the rotation of plane of polarisation of electric waves by a twisted structure,” Proc. R. Soc. London, Ser. A 63, 146–152 (1898).
    [CrossRef]
  19. O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013).
    [CrossRef]
  20. F. Brochard and P. G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. 31, 691–708 (1970).
    [CrossRef]
  21. K. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
    [CrossRef]
  22. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. II (McGraw-Hill, 1953).
  23. S. H. Schot, “Eighty years of Sommerfeld’s radiation condition,” Hist. Math. 19, 385–401 (1992).
  24. A. Rubinowicz, “A weaker formulation of the electromagnetic radiation conditions,” Rep. Math. Phys. 2, 63–77 (1971).
    [CrossRef]
  25. G. W. Ford and S. A. Werner, “Scattering and absorption of electromagnetic waves by a gyrotropic sphere,” Phys. Rev. B 18, 6752–6769 (1978).
  26. J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012).
    [CrossRef]
  27. J. Van Bladel, Electromagnetic Fields (Hemisphere, 1985).
  28. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, eds., Field Representations and Introduction to Scattering (North-Holland, 1991).
  29. C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
    [CrossRef]
  30. C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).
  31. H. Frohlich, Theory of Dielectrics (Oxford University Press, 1958).
  32. L. Ward, The Optical Constants of Bulk Materials and Films (Adam Hilger, 1988).
  33. A. Lakhtakia, “Rayleigh scattering by a bianisotropic ellipsoid in a biisotropic medium,” Int. J. Electron. 71, 1057–1062 (1991).
    [CrossRef]
  34. V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
    [CrossRef]

2013

O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013).
[CrossRef]

2012

J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012).
[CrossRef]

A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012).
[CrossRef]

1992

S. H. Schot, “Eighty years of Sommerfeld’s radiation condition,” Hist. Math. 19, 385–401 (1992).

1991

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991).
[CrossRef]

A. Lakhtakia, “Rayleigh scattering by a bianisotropic ellipsoid in a biisotropic medium,” Int. J. Electron. 71, 1057–1062 (1991).
[CrossRef]

1988

C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).

1987

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

1986

K. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
[CrossRef]

1978

G. W. Ford and S. A. Werner, “Scattering and absorption of electromagnetic waves by a gyrotropic sphere,” Phys. Rev. B 18, 6752–6769 (1978).

1974

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

1971

A. Rubinowicz, “A weaker formulation of the electromagnetic radiation conditions,” Rep. Math. Phys. 2, 63–77 (1971).
[CrossRef]

1970

F. Brochard and P. G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. 31, 691–708 (1970).
[CrossRef]

1963

V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
[CrossRef]

1908

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Lpz. 25, 377–445 (1908).

1898

J. C. Bose, “On the rotation of plane of polarisation of electric waves by a twisted structure,” Proc. R. Soc. London, Ser. A 63, 146–152 (1898).
[CrossRef]

1890

L. V. Lorenz, “Lysvevægelsen i og uden for en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Forh. 6, 1–62 (1890).

1857

M. Faraday, “Experimental relations of gold (and other materials) to light,” Philos. Trans. R. Soc. London 147, 145–181 (1857).
[CrossRef]

1850

O. F. Mossotti, “Discussione analitica sull influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricita alla superficie di più corpi elettrici disseminati in esso,” Mem. Mat. Fis. Modena 24, 49–74 (1850).

1847

O. F. Mossotti, “Recherches théoriques sur l’induction électro-statique, envisagée d’après idées de Faraday,” Supp. Biblio. Univer. Genève Arch. Sci. Phys. Natur. 16, 193–198 (1847).

Aydin, K.

K. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
[CrossRef]

Bohren, C. F.

C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

Bose, J. C.

J. C. Bose, “On the rotation of plane of polarisation of electric waves by a twisted structure,” Proc. R. Soc. London, Ser. A 63, 146–152 (1898).
[CrossRef]

Brochard, F.

F. Brochard and P. G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. 31, 691–708 (1970).
[CrossRef]

Campbell, T. A.

O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013).
[CrossRef]

de Gennes, P. G.

F. Brochard and P. G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. 31, 691–708 (1970).
[CrossRef]

Faraday, M.

M. Faraday, “Experimental relations of gold (and other materials) to light,” Philos. Trans. R. Soc. London 147, 145–181 (1857).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. II (McGraw-Hill, 1953).

Ford, G. W.

G. W. Ford and S. A. Werner, “Scattering and absorption of electromagnetic waves by a gyrotropic sphere,” Phys. Rev. B 18, 6752–6769 (1978).

Frohlich, H.

H. Frohlich, Theory of Dielectrics (Oxford University Press, 1958).

Gribble, C. D.

C. D. Gribble and A. J. Hall, Optical Mineralogy, Principles and Practice (University College London Press, 1992).

Hall, A. J.

C. D. Gribble and A. J. Hall, Optical Mineralogy, Principles and Practice (University College London Press, 1992).

Hizal, A.

K. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

Ivanova, O. S.

O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012).
[CrossRef]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991).
[CrossRef]

A. Lakhtakia, “Rayleigh scattering by a bianisotropic ellipsoid in a biisotropic medium,” Int. J. Electron. 71, 1057–1062 (1991).
[CrossRef]

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).

Li, J. L.-W.

J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012).
[CrossRef]

Lorenz, L. V.

L. V. Lorenz, “Lysvevægelsen i og uden for en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Forh. 6, 1–62 (1890).

Mackay, T. G.

A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012).
[CrossRef]

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Lpz. 25, 377–445 (1908).

G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions—translation,” Sandia Laboratories, Albuquerque, New Mexico, 1978, SAND78-6018. National Translation Center, Chicago, Illinois, Translation 79–21946.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. II (McGraw-Hill, 1953).

Mossotti, O. F.

O. F. Mossotti, “Discussione analitica sull influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricita alla superficie di più corpi elettrici disseminati in esso,” Mem. Mat. Fis. Modena 24, 49–74 (1850).

O. F. Mossotti, “Recherches théoriques sur l’induction électro-statique, envisagée d’après idées de Faraday,” Supp. Biblio. Univer. Genève Arch. Sci. Phys. Natur. 16, 193–198 (1847).

Ong, W.-L.

J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “A weaker formulation of the electromagnetic radiation conditions,” Rep. Math. Phys. 2, 63–77 (1971).
[CrossRef]

Schot, S. H.

S. H. Schot, “Eighty years of Sommerfeld’s radiation condition,” Hist. Math. 19, 385–401 (1992).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, 1941), Chap. 7.

Van Bladel, J.

J. Van Bladel, Electromagnetic Fields (Hemisphere, 1985).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991).
[CrossRef]

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991).
[CrossRef]

Walther, A.

A. Walther, “Optical applications of solid glass spheres,” Ph.D. thesis (Delft University of Technology, 1959).

Ward, L.

L. Ward, The Optical Constants of Bulk Materials and Films (Adam Hilger, 1988).

Werner, S. A.

G. W. Ford and S. A. Werner, “Scattering and absorption of electromagnetic waves by a gyrotropic sphere,” Phys. Rev. B 18, 6752–6769 (1978).

Weston, V. H.

V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
[CrossRef]

Williams, C. B.

O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013).
[CrossRef]

Zheng, K. H. R.

J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012).
[CrossRef]

Am. J. Phys.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

Ann. Phys. Lpz.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. Lpz. 25, 377–445 (1908).

Chem. Phys. Lett.

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

Hist. Math.

S. H. Schot, “Eighty years of Sommerfeld’s radiation condition,” Hist. Math. 19, 385–401 (1992).

IEEE Trans. Antennas Propag.

V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
[CrossRef]

Int. J. Electron.

A. Lakhtakia, “Rayleigh scattering by a bianisotropic ellipsoid in a biisotropic medium,” Int. J. Electron. 71, 1057–1062 (1991).
[CrossRef]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Plane waves and canonical sources in a gyroelectromagnetic uniaxial medium,” Int. J. Electron. 71, 853–861 (1991).
[CrossRef]

J. Math. Anal. Appl.

K. Aydin and A. Hizal, “On the completeness of the spherical vector wave functions,” J. Math. Anal. Appl. 117, 428–440 (1986).
[CrossRef]

J. Opt.

A. Lakhtakia and T. G. Mackay, “Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties,” J. Opt. 41, 201–213 (2012).
[CrossRef]

J. Phys.

F. Brochard and P. G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. 31, 691–708 (1970).
[CrossRef]

K. Dan. Vidensk. Selsk. Forh.

L. V. Lorenz, “Lysvevægelsen i og uden for en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Forh. 6, 1–62 (1890).

Mem. Mat. Fis. Modena

O. F. Mossotti, “Discussione analitica sull influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricita alla superficie di più corpi elettrici disseminati in esso,” Mem. Mat. Fis. Modena 24, 49–74 (1850).

Philos. Trans. R. Soc. London

M. Faraday, “Experimental relations of gold (and other materials) to light,” Philos. Trans. R. Soc. London 147, 145–181 (1857).
[CrossRef]

Phys. Rev. B

G. W. Ford and S. A. Werner, “Scattering and absorption of electromagnetic waves by a gyrotropic sphere,” Phys. Rev. B 18, 6752–6769 (1978).

Phys. Rev. E

J. L.-W. Li, W.-L. Ong, and K. H. R. Zheng, “Anisotropic scattering effects of a gyrotropic sphere characterized using the T-matrix method,” Phys. Rev. E 85, 036601 (2012).
[CrossRef]

Pigment Cell Res.

C. F. Bohren, “Understanding colors in nature,” Pigment Cell Res. 1, 214–222 (1988).

Proc. R. Soc. London, Ser. A

J. C. Bose, “On the rotation of plane of polarisation of electric waves by a twisted structure,” Proc. R. Soc. London, Ser. A 63, 146–152 (1898).
[CrossRef]

Rapid Prototyping

O. S. Ivanova, C. B. Williams, and T. A. Campbell, “Additive manufacturing (AM) and nanotechnology: promises and challenges,” Rapid Prototyping 19, 353–364 (2013).
[CrossRef]

Rep. Math. Phys.

A. Rubinowicz, “A weaker formulation of the electromagnetic radiation conditions,” Rep. Math. Phys. 2, 63–77 (1971).
[CrossRef]

Supp. Biblio. Univer. Genève Arch. Sci. Phys. Natur.

O. F. Mossotti, “Recherches théoriques sur l’induction électro-statique, envisagée d’après idées de Faraday,” Supp. Biblio. Univer. Genève Arch. Sci. Phys. Natur. 16, 193–198 (1847).

Other

T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide (World Scientific, 2010).

C. D. Gribble and A. J. Hall, Optical Mineralogy, Principles and Practice (University College London Press, 1992).

A. Walther, “Optical applications of solid glass spheres,” Ph.D. thesis (Delft University of Technology, 1959).

“Spherical glass solar energy generator by andre rawlemon,” August25, 2012, http://www.designboom.com/technology/spherical-glass-solar-energy-generator-by-rawlemon/ (accessed on June 3, 2013).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

M. Kerker, ed., Selected Papers on Light Scattering, Part 1 (SPIE, 1988).

G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions—translation,” Sandia Laboratories, Albuquerque, New Mexico, 1978, SAND78-6018. National Translation Center, Chicago, Illinois, Translation 79–21946.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, 1941), Chap. 7.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), Chap. 4.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. II (McGraw-Hill, 1953).

J. Van Bladel, Electromagnetic Fields (Hemisphere, 1985).

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, eds., Field Representations and Introduction to Scattering (North-Holland, 1991).

H. Frohlich, Theory of Dielectrics (Oxford University Press, 1958).

L. Ward, The Optical Constants of Bulk Materials and Films (Adam Hilger, 1988).

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Figures (11)

Fig. 1.
Fig. 1.

Total scattering efficiency—computed exactly using Eq. (20) and approximately using Eq. (66)—as a function of k0a, when ϵr=4 and μr=1.1. (a) Einc=x^exp(ik0z)Vm1 and αx=αy=1.2. (b) Einc=x^exp(ik0z)Vm1, αx=1.1, and αy=1.2.

Fig. 2.
Fig. 2.

Total scattering efficiency—computed exactly using Eq. (20) and approximately using Eq. (66)—as a function of k0a, when ϵr=4, μr=1.1, αx=1.1, and αy=1.2. (a) Einc=x^exp(ik0y)Vm1, (b) Einc=z^exp(ik0y)Vm1.

Fig. 3.
Fig. 3.

Differential scattering efficiency as a function of θ, when Einc=x^exp(ik0z)Vm1, ϵr=4, μr=1.1, αx=1.1, and αy=1.2. The dashed red line is for k0a=0.5, the solid blue line is for k0a=2.5, and the dashed-and-dotted black line is for k0a=4.5. (a) ϕ=0°, (b) ϕ=90°.

Fig. 4.
Fig. 4.

Same as Fig. 3, except that Einc=y^exp(ik0z)Vm1.

Fig. 5.
Fig. 5.

Differential scattering efficiency as a function of θ, when Einc=x^exp(ik0z)Vm1, k0a=4.5, ϵr=4, and μr=1.1. The red dashed line is for αx=αy=1.1, and the solid blue line is for αx=1.1 and αy=1.2. (a) ϕ=0°, (b) ϕ=90°.

Fig. 6.
Fig. 6.

Same as Fig. 5, except that Einc=y^exp(ik0z)Vm1.

Fig. 7.
Fig. 7.

Extinction, total scattering, absorption, backscattering, and forward-scattering efficiencies as functions of k0a, when k^incz^, ϵr=4(1+i0.1), μr=1.1, αx=1.1, and αy=1.2. (a) e^incx^, (b) e^incy^.

Fig. 8.
Fig. 8.

Same as Fig. 7, except that k^incx^. (a) e^incy^, (b) e^incz^.

Fig. 9.
Fig. 9.

Same as Fig. 7, except that k^incy^. (a) e^incx^, (b) e^incz^.

Fig. 10.
Fig. 10.

Backscattering efficiency as a function of k0a, when ϵr=4(1+i0.1), μr=1.1, αx=1.1, and αy=1.2, for six different combinations of e^inc and k^inc.

Fig. 11.
Fig. 11.

Same as Fig. 10, except that ϵr=μr=2(1+i0.1), αx=1.1, and αy=1.2.

Equations (67)

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

D(r,ω)=ϵ0ϵr(ω)(ω)·(ω)·E(r,ω)B(r,ω)=μ0μr(ω)(ω)·(ω)·H(r,ω)},
(ω)=αx1(ω)x^x^+αy1(ω)y^y^+z^z^,
Einc(r)=s{e,o}n=1m=0n{Dmn[Asmn(1)Msmn(1)(k0r)+Bsmn(1)Nsmn(1)(k0r)]},r<b,
Hinc(r)=k0iωμ0s{e,o}n=1m=0n{Dmn[Asmn(1)Nsmn(1)(k0r)+Bsmn(1)Msmn(1)(k0r)]},r<b.
Memn(1)(k0r)=×[rjn(k0r)Pnm(cosθ)cos(mϕ)]Momn(1)(k0r)=×[rjn(k0r)Pnm(cosθ)sin(mϕ)]Nsmn(1)(k0r)=k01×Msmn(1)(k0r),s{e,o}},m{0,1,2,,n},n{1,2,3,}
Dmn=(2δm0)(2n+1)(nm)!4n(n+1)(n+m)!
kinc=k0(x^sinθinccosϕinc+y^sinθincsinϕinc+z^cosθinc)
Einc(r)=eincexp(ikinc·r)
Hinc(r)=kinc×eincωμ0exp(ikinc·r).
eincexp(ikinc·r)=4s{e,o}n=1m=0n{inDnmn(n+1)[Csmn(θinc,ϕinc)Msmn(1)(k0r)iBsmn(θinc,ϕinc)Nsmn(1)(k0r)]},
Bemn(θ,ϕ)=1n(n+1)[dPnm(cosθ)dθcos(mϕ)θ^mPnm(cosθ)sinθsin(mϕ)ϕ^]Bomn(θ,ϕ)=1n(n+1)[dPnm(cosθ)dθsin(mϕ)θ^+mPnm(cosθ)sinθcos(mϕ)ϕ^]Cemn(θ,ϕ)=1n(n+1)[mPnm(cosθ)sinθsin(mϕ)θ^+dPnm(cosθ)dθcos(mϕ)ϕ^]Comn(θ,ϕ)=1n(n+1)[mPnm(cosθ)sinθcos(mϕ)θ^dPnm(cosθ)dθsin(mϕ)ϕ^]},
Asmn(1)=4inn(n+1)einc·Csmn(θinc,ϕinc)Bsmn(1)=4in1n(n+1)einc·Bsmn(θinc,ϕinc)}.
Esca(r)=s{e,o}n=1m=0n{Dmn[Asmn(3)Msmn(3)(k0r)+Bsmn(3)Nsmn(3)(k0r)]},r>a,
Hsca(r)=k0iωμ0s{e,o}n=1m=0n{Dmn[Asmn(3)Nsmn(3)(k0r)+Bsmn(3)Msmn(3)(k0r)]},r>a,
Memn(3)(k0r)=×[rhn(1)(k0r)Pnm(cosθ)cos(mϕ)]Momn(3)(k0r)=×[rhn(1)(k0r)Pnm(cosθ)sin(mϕ)]Nsmn(3)(k0r)=k01×Msmn(3)(k0r),s{e,o}},m{0,1,2,,n},n{1,2,3,},
limk0r[k0rexp(ik0r)hn(1)(k0r)]=(i)n+1
limk0r[k0rexp(ik0r)Msmn(3)(k0r)]=(i)n+1n(n+1)Csmn(θ,ϕ)limk0r[k0rexp(ik0r)Nsmn(3)(k0r)]=(i)nn(n+1)Bsmn(θ,ϕ)}.
Esca(r)Fsca(θ,ϕ)exp(ik0r)r
Hsca(r)η01r^×Fsca(θ,ϕ)exp(ik0r)r,
Fsca(θ,ϕ)=k01s{e,o}n=1m=0n{(i)nDmnn(n+1)[iAsmn(3)Csmn(θ,ϕ)+Bsmn(3)Bsmn(θ,ϕ)]}.
limk0r|rEsca(r)ik0Esca(r)|O(r2)limk0r|rHsca(r)ik0Hsca(r)|O(r2)}
limk0r|Esca(r)+η0r^×Hsca(r)|O(r2)limk0r|η0Hsca(r)r^×Esca(r)|O(r2)}
Eexc(r)=s{e,o}n=1m=0n[bsmnMsmn(r)+csmnNsmn(r)],r<a,
Hexc(r)=k0iωμ0ϵrμrs{e,o}n=1m=0n[bsmnNsmn(r)+csmnMsmn(r)],r<a,
Msmn(r)=Jn(kr)f1(ϕ)1·{r^[f4(ϕ)f12(ϕ)f2(θ,ϕ)sinθcosθQsmn(θ,ϕ)(αxαy)sinθsinϕcosϕRsmn(θ,ϕ)]+θ^[f4(ϕ)cos2θ+f12(ϕ)sin2θf2(θ,ϕ)Qsmn(θ,ϕ)(αxαy)cosθsinϕcosϕRsmn(θ,ϕ)]+ϕ^[αxαyf2(θ,ϕ)cosθsinϕcosϕQsmn(θ,ϕ)f4(ϕ)Rsmn(θ,ϕ)]}
Nsmn(r)=1·(r^{Jn(kr)kr[cos2θ+f4(ϕ)sin2θf22(θ,ϕ)]Psmn(θ,ϕ)+Kn(kr)f1(ϕ)[f4(ϕ)f12(ϕ)f2(θ,ϕ)sinθcosθRsmn(θ,ϕ)+(αxαy)sinθsinϕcosϕQsmn(θ,ϕ)]}+θ^{Jn(kr)kr[f4(ϕ)1f22(θ,ϕ)sinθcosθ]Psmn(θ,ϕ)+Kn(kr)f1(ϕ)[f4(ϕ)cos2θ+f12(ϕ)sin2θf2(θ,ϕ)Rsmn(θ,ϕ)+(αxαy)cosθsinϕcosϕQsmn(θ,ϕ)]}+ϕ^{Jn(kr)kr[αxαyf22(θ,ϕ)sinθsinϕcosϕ]Psmn(θ,ϕ)+Kn(kr)f1(ϕ)[αxαyf2(θ,ϕ)cosθsinϕcosϕRsmn(θ,ϕ)+f4(ϕ)Qsmn(θ,ϕ)]}),
Jn(kr)=jn[krf2(θ,ϕ)],
Kn(kr)=n+1krf2(θ,ϕ)Jn(kr)Jn+1(kr),
Psmn(θ,ϕ)=n(n+1)Pnm[cosθf2(θ,ϕ)]Vsm(ϕ),
Qsmn(θ,ϕ)=mPnm[cosθf2(θ,ϕ)]f2(θ,ϕ)f1(ϕ)sinθUsm(ϕ),
Rsmn(θ,ϕ)=1f1(ϕ)sinθ{(nm+1)f2(θ,ϕ)Pn+1m[cosθf2(θ,ϕ)](n+1)cosθPnm[cosθf2(θ,ϕ)]}Vsm(ϕ),
Usm(ϕ)={sin[mf3(ϕ)]cos[mf3(ϕ)]},s={eo,
Vsm(ϕ)={cos[mf3(ϕ)]sin[mf3(ϕ)]},s={eo,
f1(ϕ)=+(αx2cos2ϕ+αy2sin2ϕ)1/2,
f2(θ,ϕ)=+[f12(ϕ)sin2θ+cos2θ]1/2,
f3(ϕ)=tan1(αyαxtanϕ),
f4(ϕ)=αxcos2ϕ+αysin2ϕ.
θ^·Eexc(r)=θ^·[Einc(r)+Esca(r)]ϕ^·Eexc(r)=ϕ^·[Einc(r)+Esca(r)]θ^·Hexc(r)=θ^·[Hinc(r)+Hsca(r)]ϕ^·Hexc(r)=ϕ^·[Hinc(r)+Hsca(r)]},r=a,
Asmn(1)=s{e,o}n=1m=0n[Ismn,smn(1)bsmn+Jsmn,smn(1)csmn],
Bsmn(1)=s{e,o}n=1m=0n[Ksmn,smn(1)bsmn+Lsmn,smn(1)csmn],
Asmn(3)=s{e,o}n=1m=0n[Ismn,smn(3)bsmn+Jsmn,smn(3)csmn],
Bsmn(3)=s{e,o}n=1m=0n[Ksmn,smn(3)bsmn+Lsmn,smn(3)csmn].
Ismn,smn(j)=i(k0a)2π02πdϕ0πdθsinθ{Nsmn()(k0ar^)·[r^×Msmn(ar^)]+ϵrμrMsmn()(k0ar^)·[r^×Nsmn(ar^)]},
Jsmn,smn(j)=i(k0a)2π02πdϕ0πdθsinθ{Nsmn()(k0ar^)·[r^×Nsmn(ar^)]+ϵrμrMsmn()(k0ar^)·[r^×Msmn(ar^)]},
Ksmn,smn(j)=i(k0a)2π02πdϕ0πdθsinθ{Msmn()(k0ar^)·[r^×Msmn(ar^)]+ϵrμrNsmn()(k0ar^)·[r^×Nsmn(ar^)]},
Lsmn,smn(j)=i(k0a)2π02πdϕ0πdθsinθ{Msmn()(k0ar^)·[r^×Nsmn(ar^)]+ϵrμrNsmn()(k0ar^)·[r^×Msmn(ar^)]},
[A(1)B(1)]=[Y(1)][bc],[A(3)B(3)]=[Y(3)][bc],
[A(3)B(3)]=[Y(3)][Y(1)]1[A(1)B(1)]=[T][A(1)B(1)].
Ismn,smn(j)δss(δmm+δm,m1+δm,m+1),
Jsmn,smn(j)(1δss)(δmm+δm,m1+δm,m+1),
Ksmn,smn(j)(1δss)(δmm+δm,m1+δm,m+1),
Lsmn,smn(j)δss(δmm+δm,m1+δm,m+1).
Ismn,smn(j)δssδmmδnn,
Jsmn,smn(j)(1δss)δmmδnn,
Ksmn,smn(j)(1δss)δmmδnn,
Lsmn,smn(j)δssδmmδnn,
σD(θ,ϕ)=4πFsca(θ,ϕ)·Fsca*(θ,ϕ)einc·einc*
σsca=1einc·einc*ϕ=02πθ=0π[Fsca(θ,ϕ)·Fsca*(θ,ϕ)]sinθdθdϕ,
=1einc·einc*πk02s{e,o}n=1m=0n[Dmn(|Asmn(3)|2+|Bsmn(3)|2)].
σext=4πk0Im[Fsca(θinc,ϕinc)·einc*einc·einc*]
s{e,o}n=1m=0ns{e,o}n=1Nm=0n
Qb=σD(π+θinc,π+ϕinc)/πa2
Qf=σD(θinc,ϕinc)/πa2,
peqvt=4πa3ϵ0(ϵr·)·(ϵr·+2)1·einc
meqvt=4πa3μ0(μr·)·(μr·+2)1·(kinc×eincωμ0),
FscaRayleigh(r^)=ω2μ04π[r^×(r^×peqvt)+ϵ0μ0r^×meqvt]=k02a3(r^×)·[(r^×)·(ϵr·)·(ϵr·+2)1+(μr·)·(μr·+2)1·(k^inc×)]·einc,
k0aπ/5k0aϵrμrmax{αx1,αy1,1}π/5},

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