Abstract

Consider that an incident plane wave is scattered by a homogeneous and isotropic magnetic sphere of finite radius. We determine, by means of the rigorous Mie theory, an exact expression for the time-averaged electromagnetic energy within this particle. For magnetic scatterers, we find that the value of the average internal energy in the resonance picks is much larger than the one associated with a scatterer with the same nonmagnetic medium properties. This result is valid even, and especially, for low size parameter values. Expressions for the contributions of the radial and angular field components to the internal energy are determined. For the analytical study of the weak absorption regime, we derive an exact expression for the absorption cross section in terms of the magnetic Mie internal coefficients. We stress that, although the electromagnetic scattering by particles is a well-documented topic, almost no attention has been devoted to magnetic scatterers. Our aim is to provide some new analytical results, which can be used for magnetic particles, and emphasize the unusual properties of the magnetic scatters, which could be important in some applications.

© 2010 Optical Society of America

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References

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  1. M. Kerker, D. S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73, 765-767 (1983).
    [CrossRef]
  2. F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435-1438 (2000).
    [CrossRef] [PubMed]
  3. F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563-5566 (2000).
    [CrossRef]
  4. F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Multiple scattering of electromagnetic waves in disordered magnetic media: localization parameter, energy transport velocity and diffusion constant,” Braz. J. Phys. 31, 65-70 (2001).
    [CrossRef]
  5. F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Electromagnetic scattering by small magnetic particles,” J. Magn. Magn. Mater. 226-230, 1951-1953 (2001).
    [CrossRef]
  6. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  7. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1980).
  8. P. W. Barber, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
    [CrossRef]
  9. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  10. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  11. M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Effect of the magnetic permeability on photonic band gaps,” Phys. Rev. B 56, 959-962 (1997).
    [CrossRef]
  12. P. Chen, R. X. Wu, J. Xu, A. M. Jiang, and X. Y. Ji, “Effects of magnetic anisotropy on the stop band of ferromagnetic electromagnetic band gap materials,” J. Phys. Condens. Matter 19, 106205 (2007).
    [CrossRef]
  13. Z. F. Lin and S. T. Chui, “Manipulating electromagnetic radiation with magnetic photonic crystals,” Opt. Lett. 32, 2288-2290 (2007).
    [CrossRef] [PubMed]
  14. S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
    [CrossRef]
  15. Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particles,” Phys. Rev. E 69, 056614 (2004).
    [CrossRef]
  16. R.-J. Tarento, K.-H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
    [CrossRef]
  17. A. Bott and W. Zdunkowski, “Electromagnetic energy within dielectric spheres,” J. Opt. Soc. Am. A 4, 1361-1365 (1987).
    [CrossRef]
  18. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  19. D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, 1999).
  20. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Mathematical Library, 1958).
  21. G. B. Arfken and H. J. Weber, Essentials of Math Methods for Physicists (Academic, 2003).
  22. P. Chyýlek, J. D. Pendleton, and R. G. Pinnick, “Internal and near-surface scattered field of a spherical particle at resonant conditions,” Appl. Opt. 24, 3940-3942 (1985).
    [CrossRef]
  23. D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Time dependence of internal intensity of a dielectric sphere on and near resonance,” J. Opt. Soc. Am. A 9, 1364-1373 (1992).
    [CrossRef]
  24. M. I. Mishchenko and A. A. Lacis, “Manifestations of morphology-dependent resonances in Mie scattering matrices,” Appl. Math. Comput. 116, 167-179 (2000).
    [CrossRef]
  25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

2008 (1)

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

2007 (2)

P. Chen, R. X. Wu, J. Xu, A. M. Jiang, and X. Y. Ji, “Effects of magnetic anisotropy on the stop band of ferromagnetic electromagnetic band gap materials,” J. Phys. Condens. Matter 19, 106205 (2007).
[CrossRef]

Z. F. Lin and S. T. Chui, “Manipulating electromagnetic radiation with magnetic photonic crystals,” Opt. Lett. 32, 2288-2290 (2007).
[CrossRef] [PubMed]

2004 (2)

Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particles,” Phys. Rev. E 69, 056614 (2004).
[CrossRef]

R.-J. Tarento, K.-H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
[CrossRef]

2003 (1)

G. B. Arfken and H. J. Weber, Essentials of Math Methods for Physicists (Academic, 2003).

2001 (2)

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Multiple scattering of electromagnetic waves in disordered magnetic media: localization parameter, energy transport velocity and diffusion constant,” Braz. J. Phys. 31, 65-70 (2001).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Electromagnetic scattering by small magnetic particles,” J. Magn. Magn. Mater. 226-230, 1951-1953 (2001).
[CrossRef]

2000 (3)

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435-1438 (2000).
[CrossRef] [PubMed]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563-5566 (2000).
[CrossRef]

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

1999 (1)

D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, 1999).

1997 (1)

M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Effect of the magnetic permeability on photonic band gaps,” Phys. Rev. B 56, 959-962 (1997).
[CrossRef]

1992 (1)

1990 (1)

P. W. Barber, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

1987 (1)

1985 (1)

1983 (2)

M. Kerker, D. S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73, 765-767 (1983).
[CrossRef]

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

1980 (1)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1980).

1978 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

1970 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

1969 (1)

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

1958 (1)

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Mathematical Library, 1958).

1941 (1)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Arfken, G. B.

G. B. Arfken and H. J. Weber, Essentials of Math Methods for Physicists (Academic, 2003).

Barber, P. W.

Bennemann, K. -H.

R.-J. Tarento, K.-H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
[CrossRef]

Biswas, R.

M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Effect of the magnetic permeability on photonic band gaps,” Phys. Rev. B 56, 959-962 (1997).
[CrossRef]

Bohren, C. F.

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

Bott, A.

Chen, P.

P. Chen, R. X. Wu, J. Xu, A. M. Jiang, and X. Y. Ji, “Effects of magnetic anisotropy on the stop band of ferromagnetic electromagnetic band gap materials,” J. Phys. Condens. Matter 19, 106205 (2007).
[CrossRef]

Chowdhury, D. Q.

Chui, S. T.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

Z. F. Lin and S. T. Chui, “Manipulating electromagnetic radiation with magnetic photonic crystals,” Opt. Lett. 32, 2288-2290 (2007).
[CrossRef] [PubMed]

Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particles,” Phys. Rev. E 69, 056614 (2004).
[CrossRef]

Chyýlek, P.

Du, J.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

Giles, C. L.

Griffiths, D. J.

D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, 1999).

Hill, S. C.

Ho, K. M.

M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Effect of the magnetic permeability on photonic band gaps,” Phys. Rev. B 56, 959-962 (1997).
[CrossRef]

Huffman, D. R.

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Ji, X. Y.

P. Chen, R. X. Wu, J. Xu, A. M. Jiang, and X. Y. Ji, “Effects of magnetic anisotropy on the stop band of ferromagnetic electromagnetic band gap materials,” J. Phys. Condens. Matter 19, 106205 (2007).
[CrossRef]

Jiang, A. M.

P. Chen, R. X. Wu, J. Xu, A. M. Jiang, and X. Y. Ji, “Effects of magnetic anisotropy on the stop band of ferromagnetic electromagnetic band gap materials,” J. Phys. Condens. Matter 19, 106205 (2007).
[CrossRef]

Joyes, P.

R.-J. Tarento, K.-H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
[CrossRef]

Kerker, M.

Lacis, A. A.

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

Lin, Z.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

Lin, Z. F.

Z. F. Lin and S. T. Chui, “Manipulating electromagnetic radiation with magnetic photonic crystals,” Opt. Lett. 32, 2288-2290 (2007).
[CrossRef] [PubMed]

Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particles,” Phys. Rev. E 69, 056614 (2004).
[CrossRef]

Liu, S.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

Martinez, A. S.

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Electromagnetic scattering by small magnetic particles,” J. Magn. Magn. Mater. 226-230, 1951-1953 (2001).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Multiple scattering of electromagnetic waves in disordered magnetic media: localization parameter, energy transport velocity and diffusion constant,” Braz. J. Phys. 31, 65-70 (2001).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563-5566 (2000).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435-1438 (2000).
[CrossRef] [PubMed]

Mishchenko, M. I.

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

Pendleton, J. D.

Pinheiro, F. A.

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Multiple scattering of electromagnetic waves in disordered magnetic media: localization parameter, energy transport velocity and diffusion constant,” Braz. J. Phys. 31, 65-70 (2001).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Electromagnetic scattering by small magnetic particles,” J. Magn. Magn. Mater. 226-230, 1951-1953 (2001).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435-1438 (2000).
[CrossRef] [PubMed]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563-5566 (2000).
[CrossRef]

Pinnick, R. G.

Sampaio, L. C.

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Multiple scattering of electromagnetic waves in disordered magnetic media: localization parameter, energy transport velocity and diffusion constant,” Braz. J. Phys. 31, 65-70 (2001).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Electromagnetic scattering by small magnetic particles,” J. Magn. Magn. Mater. 226-230, 1951-1953 (2001).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563-5566 (2000).
[CrossRef]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435-1438 (2000).
[CrossRef] [PubMed]

Sigalas, M. M.

M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Effect of the magnetic permeability on photonic band gaps,” Phys. Rev. B 56, 959-962 (1997).
[CrossRef]

Soukoulis, C. M.

M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Effect of the magnetic permeability on photonic band gaps,” Phys. Rev. B 56, 959-962 (1997).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tarento, R. -J.

R.-J. Tarento, K.-H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1980).

Van de Walle, J.

R.-J. Tarento, K.-H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
[CrossRef]

Wang, D. S.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Mathematical Library, 1958).

Weber, H. J.

G. B. Arfken and H. J. Weber, Essentials of Math Methods for Physicists (Academic, 2003).

Wu, R. X.

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

P. Chen, R. X. Wu, J. Xu, A. M. Jiang, and X. Y. Ji, “Effects of magnetic anisotropy on the stop band of ferromagnetic electromagnetic band gap materials,” J. Phys. Condens. Matter 19, 106205 (2007).
[CrossRef]

Xu, J.

P. Chen, R. X. Wu, J. Xu, A. M. Jiang, and X. Y. Ji, “Effects of magnetic anisotropy on the stop band of ferromagnetic electromagnetic band gap materials,” J. Phys. Condens. Matter 19, 106205 (2007).
[CrossRef]

Zdunkowski, W.

Appl. Math. Comput. (1)

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

Appl. Opt. (1)

Braz. J. Phys. (1)

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Multiple scattering of electromagnetic waves in disordered magnetic media: localization parameter, energy transport velocity and diffusion constant,” Braz. J. Phys. 31, 65-70 (2001).
[CrossRef]

J. Magn. Magn. Mater. (1)

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Electromagnetic scattering by small magnetic particles,” J. Magn. Magn. Mater. 226-230, 1951-1953 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. Condens. Matter (1)

P. Chen, R. X. Wu, J. Xu, A. M. Jiang, and X. Y. Ji, “Effects of magnetic anisotropy on the stop band of ferromagnetic electromagnetic band gap materials,” J. Phys. Condens. Matter 19, 106205 (2007).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

S. Liu, J. Du, Z. Lin, R. X. Wu, and S. T. Chui, “Formation of robust and completely tunable resonant photonic band gaps,” Phys. Rev. B 78, 155101 (2008).
[CrossRef]

M. M. Sigalas, C. M. Soukoulis, R. Biswas, and K. M. Ho, “Effect of the magnetic permeability on photonic band gaps,” Phys. Rev. B 56, 959-962 (1997).
[CrossRef]

Phys. Rev. E (2)

Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particles,” Phys. Rev. E 69, 056614 (2004).
[CrossRef]

R.-J. Tarento, K.-H. Bennemann, P. Joyes, and J. Van de Walle, “Mie scattering of magnetic spheres,” Phys. Rev. E 69, 026606 (2004).
[CrossRef]

Phys. Rev. Lett. (2)

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “New effects in light scattering in disordered media and coherent backscattering cone: system of magnetic particles,” Phys. Rev. Lett. 84, 1435-1438 (2000).
[CrossRef] [PubMed]

F. A. Pinheiro, A. S. Martinez, and L. C. Sampaio, “Vanishing of energy transport and diffusion constant of electromagnetic waves in disordered magnetic media,” Phys. Rev. Lett. 85, 5563-5566 (2000).
[CrossRef]

Other (10)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, 1999).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Mathematical Library, 1958).

G. B. Arfken and H. J. Weber, Essentials of Math Methods for Physicists (Academic, 2003).

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1980).

P. W. Barber, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

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Figures (5)

Fig. 1
Fig. 1

Comparison among the distributions of W ( a ) / W 0 as a function of the size parameter k a . The values of the relative permittivity and permeability related to a nonabsorptive sphere are ϵ 1 / ϵ = 1.4161 and μ 1 / μ = 1 , 10 , 100 , 1000 , respectively. The internal energy W ( a ) / W 0 is calculated in the interval 10 6 k a 1 , δ ( k a ) = 10 4 .

Fig. 2
Fig. 2

The normalized time-averaged internal energy W ( a ) / W 0 within a nonabsorptive magnetic sphere plotted as a function of the size parameter k a . The values of the relative permittivity and permeability are ϵ 1 / ϵ = 1.4161 and μ 1 / μ = 10 4 , respectively. The internal energy W ( a ) / W 0 is calculated in the interval 10 6 k a 1 , δ ( k a ) = 10 4 .

Fig. 3
Fig. 3

The normalized time-averaged internal energy W ( a ) / W 0 plotted as a function of the size parameter k a . The values of the relative permeability are μ 1 / μ = 1 (nonmagnetic sphere) and 100 (magnetic sphere). The relative refraction index is m = 1.334 + 1.5 × 10 9 ı , which have been used in [17] in the nonmagnetic scattering approach. The quantities are calculated in the interval 1 k a 50 , δ ( k a ) = 0.01 .

Fig. 4
Fig. 4

The separation of the total time-averaged internal energy in radial and angular contributions respective to both electric and magnetic fields. The values of the relative electric permittivity and magnetic permeability are ϵ 1 / ϵ = 10 and μ 1 / μ = 100 , respectively. The component contributions W r ( a ) / W 0 and W θ , ϕ ( a ) / W 0 are calculated in the interval 10 6 k a 2 , δ ( k a ) = 10 4 .

Fig. 5
Fig. 5

Ratio between the absorption efficiency Q abs and the normalized time-averaged internal energy W ( a ) / W 0 plotted as a function of the size parameter k a . The values of the relative permeability and refraction index are μ 1 / μ = 1 (nonmagnetic sphere) and m = 1.334 + 1.5 × 10 9 ı , respectively. The quantities are calculated in the interval 1 k a 49 , δ ( k a ) = 2 . The angular coefficient of linear regression is approximately 2.997 × 10 9 , which is in agreement with Eq. (29): 8 m i / ( 3 m r ) 2.998 × 10 9 .

Equations (65)

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E i ( r , t ) = E 0   exp [ ı ( k r ω t ) ] ,
E 1 r = ı   cos   ϕ   sin θ ρ 1 2 n = 1 E n d n ψ n ( ρ 1 ) n ( n + 1 ) π n ,
E 1 θ = cos   ϕ ρ 1 n = 1 E n [ c n π n ψ n ( ρ 1 ) ı d n τ n ψ n ( ρ 1 ) ] ,
E 1 ϕ = sin   ϕ ρ 1 n = 1 E n [ ı d n π n ψ n ( ρ 1 ) c n τ n ψ n ( ρ 1 ) ] ,
H 1 r = ı k 1 ω μ 1 sin   ϕ   sin   θ ρ 1 2 n = 1 E n c n ψ n ( ρ 1 ) n ( n + 1 ) π n ,
H 1 θ = k 1 ω μ 1 sin   ϕ ρ 1 n = 1 E n [ d n π n ψ n ( ρ 1 ) ı c n τ n ψ n ( ρ 1 ) ] ,
H 1 ϕ = k 1 ω μ 1 cos   ϕ ρ 1 n = 1 E n [ d n τ n ψ n ( ρ 1 ) ı c n π n ψ n ( ρ 1 ) ] ,
a n = m ̃ ψ n ( m x ) ψ n ( x ) ψ n ( x ) ψ n ( m x ) m ̃ ψ n ( m x ) ξ n ( x ) ξ n ( x ) ψ n ( m x ) ,
b n = ψ n ( m x ) ψ n ( x ) m ̃ ψ n ( x ) ψ n ( m x ) ψ n ( m x ) ξ n ( x ) m ̃ ξ n ( x ) ψ n ( m x ) ,
c n = m ı ψ n ( m x ) ξ n ( x ) m ̃ ξ n ( x ) ψ n ( m x ) ,
d n = m ı m ̃ ψ n ( m x ) ξ n ( x ) ξ n ( x ) ψ n ( m x ) ,
W ( a ) = 0 2 π d ϕ 1 1 d ( cos   θ ) 0 a d r r 2   Re [ ϵ 1 4 ( | E 1 r | 2 + | E 1 θ | 2 + | E 1 ϕ | 2 ) + μ 1 4 ( | H 1 r | 2 + | H 1 θ | 2 + | H 1 ϕ | 2 ) ] .
W r H ( a ) = π 2 | E 0 | 2 Re ( μ 1 1 ) ω 2 n = 1 n ( n + 1 ) ( 2 n + 1 ) | c n | 2 0 a d r | j n ( ρ 1 ) | 2 .
[ W θ E + W ϕ E ] ( a ) = Re ( ϵ 1 ) 4 1 1 d ( cos   θ ) 0 2 π d ϕ 0 a d r r 2 ( | E 1 θ | 2 + | E 1 ϕ | 2 ) = π 2 | E 0 | 2 Re ( ϵ 1 ) | k 1 | 2 n = 1 ( 2 n + 1 ) 0 a d r ( | c n ψ n ( ρ 1 ) | 2 + | d n ψ n ( ρ 1 ) | 2 ) .
[ W θ H + W ϕ H ] ( a ) = π 2 | E 0 | 2 Re ( μ 1 1 ) ω 2 n = 1 ( 2 n + 1 ) 0 a d r ( | d n ψ n ( ρ 1 ) | 2 + | c n ψ n ( ρ 1 ) | 2 ) .
W E ( a ) = W r E ( a ) + [ W θ E + W ϕ E ] ( a ) = 3 4 W 0   Re ( m m ̃ ) n = 1 { ( 2 n + 1 ) | c n | 2 I n ( y ) + | d n | 2 [ n I n + 1 ( y ) + ( n + 1 ) I n 1 ( y ) ] } ,
I n ( y ) = 1 a 3 0 a d r r 2 | j n ( ρ 1 ) | 2
W 0 = 2 3 π a 3 | E 0 | 2 ϵ .
W H ( a ) = W r H ( a ) + [ W θ H + W ϕ H ] ( a ) = 3 4 W 0   Re ( m m ̃ ) n = 1 { ( 2 n + 1 ) | d n | 2 I n ( y ) + | c n | 2 [ n I n + 1 ( y ) + ( n + 1 ) I n 1 ( y ) ] } .
W ( a ) = 3 4 W 0 n = 1 | ψ n ( y ) | 2 [ n β n I n + 1 ( y ) + ( n + 1 ) β n I n 1 ( y ) + ( 2 n + 1 ) α n I n ( y ) ] ,
α n = | ψ n ( y ) | 2 [ Re ( m m ̃ ) | c n | 2 + Re ( m m ̃ ) | d n | 2 ] ,
β n = | ψ n ( y ) | 2 [ Re ( m m ̃ ) | d n | 2 + Re ( m m ̃ ) | c n | 2 ] .
W ( a ) = 3 4 W 0 n = 1 2 n + 1 y 2 y 2 { α n [ A n ( y ) y A n ( y ) y ] + β n [ A n ( y ) y A n ( y ) y ] } ,
σ sca = 2 π k 2 n = 1 ( 2 n + 1 ) ( | a n | 2 + | b n | 2 ) ,
σ tot = 2 π k 2 n = 1 ( 2 n + 1 ) Re { a n + b n } .
h n ( 1 ) ( x ) b n = j n ( x ) j n ( m x ) c n ,
h n ( 1 ) ( x ) a n = j n ( x ) m ̃ j n ( m x ) d n .
σ abs = 2 π k 2 n = 1 ( 2 n + 1 ) { Re [ ψ n ( m x ) m ξ n ( x ) ( c n + m ̃ d n ) ] | ψ n ( m x ) | 2 | m ξ n ( x ) | 2 ( | c n | 2 + | m ̃ d n | 2 ) } .
σ abs = 2 π k 2 n = 1 ( 2 n + 1 ) ( | c n | 2 + | d n | 2 ) Im [ m ̃ | m | 2 ψ n ( y ) ψ n ( y ) ] .
W E norm ( m , m ̃ , k a ) = W E ( m , m ̃ , k a ; ϵ , a ) W 0 ( ϵ , a ) ,
W H norm ( m , m ̃ , k a ) = W H ( m , m ̃ , k a ; ϵ , a ) W 0 ( ϵ , a ) ,
W r E ( a ) W 0 = 3 4 Re ( m m ̃ ) n = 1 n ( n + 1 ) 2 n + 1 | d n | 2 { I n 1 ( y ) + I n + 1 ( y ) + 2 a 3 0 a d r r 2   Re [ j n 1 ( ρ 1 ) j n + 1 ( ρ 1 ) ] } .
W wa ( a ) 3 8 W 0 m r m i 2 m ̃ r x 3 m r 2 n = 1 ( 2 n + 1 ) ( | c n | 2 + | d n | 2 ) Im [ ψ n ( y ) ψ n ( y ) ] .
W wa ( a ) 3 8 W 0 m r m i x Q abs ,
0 a r 2 | j n ( ρ 1 ) | 2 d r = a 3 [ y j n ( y ) j n ( y ) y j n ( y ) j n ( y ) ] y 2 y 2 = 2 a 3 | j n ( y ) | 2 Re [ φ n ( y ) y 2 y 2 ] ,
0 a r 2 j n 2 ( ρ 1 ) d r = a 3 2 [ j n 2 ( y ) j n 1 ( y ) j n + 1 ( y ) ] ,
1 1 d ( cos   θ ) ( π n π n + τ n τ n ) = 2 n 2 ( n + 1 ) 2 2 n + 1 δ n , n ,
1 1 d ( cos   θ ) ( π n τ n + τ n π n ) = 0 ,
1 1 d ( cos   θ ) π n π n sin 2 θ = 2 n ( n + 1 ) 2 n + 1 δ n , n .
a 1 ı x 3 3 φ 1 ( m x ) 2 m m ̃ φ 1 ( m x ) + m m ̃ ı x 5 5 [ φ 1 ( m x ) m m ̃ ] 2 m m ̃ φ 1 ( m x ) [ φ 1 ( m x ) + m m ̃ ] 2 + x 6 9 [ φ 1 ( m x ) 2 m m ̃ φ 1 ( m x ) + m m ̃ ] 2 + O ( x 7 ) ,
b 1 ı x 3 3 φ 1 ( m x ) 2 m / m ̃ φ 1 ( m x ) + m / m ̃ ı x 5 5 [ φ 1 ( m x ) m / m ̃ ] 2 ( m / m ̃ ) φ 1 ( m x ) [ φ 1 ( m x ) + m / m ̃ ] 2 + x 6 9 [ φ 1 ( m x ) 2 m / m ̃ φ 1 ( m x ) + m / m ̃ ] 2 + O ( x 7 ) ,
a 2 ı x 5 45 φ 2 ( m x ) 3 m m ̃ φ 2 ( m x ) + 2 m m ̃ + O ( x 7 ) ,
b 2 ı x 5 45 φ 2 ( m x ) 3 m / m ̃ φ 2 ( m x ) + 2 m / m ̃ + O ( x 7 ) .
c 1 m x 2 ψ 1 ( m x ) m / m ̃ [ φ 1 ( m x ) + m / m ̃ ]
m x 4 ψ 1 ( m x ) ( m / m ̃ ) [ φ 1 ( m x ) m / m ̃ ] 2 [ φ 1 ( m x ) + m / m ̃ ] 2 + O ( x 5 ) ,
d 1 ( m / m ̃ ) x 2 ψ 1 ( m x ) m m ̃ [ φ 1 ( m x ) + m m ̃ ]
( m / m ̃ ) x 4 ψ 1 ( m x ) m m ̃ [ φ 1 ( m x ) m m ̃ ] 2 [ φ 1 ( m x ) + m m ̃ ] 2 + O ( x 5 ) ,
c 2 m 3 ψ 2 ( m x ) ( m / m ̃ ) x 3 [ φ 2 ( m x ) + 2 m / m ̃ ] + O ( x 5 ) ,
d 2 ( m / m ̃ ) 3 ψ 2 ( m x ) m m ̃ x 3 [ φ 2 ( m x ) + 2 m m ̃ ] + O ( x 5 ) .
a 1 2 ı x 3 3 m m ̃ 1 m m ̃ + 2 ı x 5 5 m 3 m ̃ 6 m m ̃ + ( m m ̃ ) 2 + 4 ( m m ̃ + 2 ) 2 + 4 x 6 9 ( m m ̃ 1 m m ̃ + 2 ) 2 + O ( x 7 ) ,
b 1 2 ı x 3 3 m / m ̃ 1 m / m ̃ + 2 ı x 5 5 m 3 m ̃ 6 m / m ̃ + ( m / m ̃ ) 2 + 4 ( m / m ̃ + 2 ) 2 + 4 x 6 9 ( m / m ̃ 1 m / m ̃ + 2 ) 2 + O ( x 7 ) ,
a 2 ı x 5 15 m m ̃ 1 2 m m ̃ + 3 + O ( x 7 ) ,
b 2 ı x 5 15 m / m ̃ 1 2 m / m ̃ + 3 + O ( x 7 ) ,
c 1 3 2 m ̃ + m [ 1 + ( m x ) 2 10 ] 3 x 2 2 [ 1 + ( m x ) 2 10 ] ( 2 m ̃ m ) ( 2 m ̃ + m ) 2 + O ( x 5 ) ,
d 1 3 2 + m m ̃ [ 1 + ( m x ) 2 10 ] 3 x 2 2 [ 1 + ( m x ) 2 10 ] ( 2 m m ̃ ) ( 2 + m m ̃ ) 2 + O ( x 5 ) ,
c 2 5 m m ̃ ( 3 + 2 m / m ̃ ) + O ( x 5 ) ,
d 2 5 m ( 3 + 2 m m ̃ ) + O ( x 5 ) .
a 1 ı x 3 3 x   tan ( m x ) + 2 m ̃ x   tan ( m x ) m ̃ ı x 5 5 [ x   tan ( m x ) + m ̃ ] 2 + m ̃ x   tan ( m x ) [ x   tan ( m x ) m ̃ ] 2 + x 6 9 [ x   tan ( m x ) + 2 m ̃ x   tan ( m x ) m ̃ ] 2 + O ( x 7 ) ,
b 1 ı x 3 3 m ̃ x   tan ( m x ) + 2 m ̃ x   tan ( m x ) 1 ı x 5 5 [ m ̃ x   tan ( m x ) + 1 ] 2 + m ̃ x   tan ( m x ) [ m ̃ x   tan ( m x ) 1 ] 2 + x 6 9 [ m ̃ x   tan ( m x ) + 2 m ̃ x   tan ( m x ) 1 ] 2 + O ( x 7 ) ,
a 2 ı x 5 45 x 3 m ̃   tan ( m x ) x + 2 m ̃   tan ( m x ) + O ( x 7 ) ,
b 2 ı x 5 45 m ̃ x 3   tan ( m x ) m ̃ x + 2   tan ( m x ) + O ( x 7 ) .
c 1 m cos ( m x ) x 2 [ m ̃ x   tan ( m x ) 1 ] m x 4 2   cos ( m x ) [ m ̃ x   tan ( m x ) + 1 ] [ m ̃ x   tan ( m x ) 1 ] 2 + O ( x 5 ) ,
d 1 m cos ( m x ) x 2 [ x   tan ( m x ) m ̃ ] m x 4 2   cos ( m x ) [ x   tan ( m x ) + m ̃ ] [ x   tan ( m x ) m ̃ ] 2 + O ( x 5 ) ,
c 2 m 3   cos ( m x ) x 3 [ 2   tan ( m x ) + m ̃ x ] + O ( x 5 ) ,
d 2 m 3   cos ( m x ) x 3 [ 2 m ̃   tan ( m x ) + x ] + O ( x 5 ) .

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