Abstract

Analytical expressions are obtained for the derivatives of Mie scattering coefficients with respect to the electrical radius of the spherical scattering particle, and to the relative permittivity and permeability of both the particle and the surrounding medium. Their corresponding approximate expressions are developed to avoid numerical overflow based on the logarithmic derivative of Riccati–Bessel functions. The analytical expressions have been verified by comparing their results with those calculated by analytical expressions developed by Mathematica. Compared with the numerical derivative, the analytical expressions and approximate expressions show a higher accuracy and are 2.0 and 2.8 times, respectively, faster in the case of a single magnetodielectric sphere. Generally, for spheres with an electrical radius in a large range, the approximate expressions can yield acceptable accuracy and computation time up to a high order. This work can be used in the design of nonmetallic metamaterials, and in the retrieval of aerosol properties from remote sensing data. An example calculation is given for the design of an optical, all-dielectric, mu-negative metamaterial consisting of a simple cubic array of tellurium nanoparticles.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
    [CrossRef]
  2. R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
    [CrossRef]
  3. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
    [CrossRef]
  4. Y. Li and N. Bowler, “Rational design of double-negative metamaterials consisting of 3D arrays of two different nonmetallic spheres arranged on a simple tetragonal lattice,” in 2011 IEEE International Symposium on Antennas and Propagation (2011), pp. 1494–1497.
  5. Y. Li and N. Bowler, “Analysis of double-negative (DNG) bandwidths for metamaterials composed of three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Antennas Wirel. Propag. Lett. 10, 1484–1487 (2011).
    [CrossRef]
  6. Y. Li and N. Bowler, “Traveling waves on three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Trans. Antennas Propag. 60, 2727–2739 (2012).
    [CrossRef]
  7. I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006).
    [CrossRef]
  8. I. B. Vendik, O. G. Vendik, and M. A. Odit, “An isotropic metamaterial formed with ferroelectric ceramic spherical inclusions,” Phys. Solid State 51, 1590–1594 (2009).
    [CrossRef]
  9. K. L. Kumley and E. F. Kuester, “Effect of scatterer size variations on the reflection and transmission properties of a metafilm,” presented at National Radio Science Meeting, Boulder, CO (2012).
  10. E. F. Kuester, K. L. Kumley, and C. L. Holloway, “Effect of sphere radius variation on the guided waves of a metafilm,” in 2013 IEEE International Symposium on Antennas and Propagation (2013).
  11. Y. Li and N. Bowler, “Effects of parameter variations on negative effective constitutive parameters of nonmetallic metamaterials,” J. Appl. Phys. 113, 063501 (2013).
    [CrossRef]
  12. G. Thomas, S. Bass, R. Grainger, and A. Lambert, “Retrieval of aerosol refractive index from extinction spectra with a damped harmonic-oscillator band model,” Appl. Opt. 44, 1332–1341 (2005).
    [CrossRef]
  13. O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
    [CrossRef]
  14. F. Xu and A. B. Davis, “Derivatives of light scattering properties of a nonspherical particle computed with the T-matrix method,” Opt. Lett. 36, 4464–4466 (2011).
    [CrossRef]
  15. R. Spurr, J. Wang, J. Zeng, and M. I. Mishchenko, “Linearized T-matrix and Mie scattering computations,” J. Quant. Spectrosc. Radiat. Transfer 113, 425–439 (2012).
    [CrossRef]
  16. R. Grainger, J. Lucas, G. Thomas, and G. Ewen, “Calculation of Mie derivatives,” Appl. Opt. 43, 5386–5393 (2004).
    [CrossRef]
  17. G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 330, 377–445 (1908).
    [CrossRef]
  18. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).
  20. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).
  22. B. Verner, “Note on the recurrence between Mie’s coefficients,” J. Opt. Soc. Am. 66, 1424–1425 (1976).
    [CrossRef]
  23. H. Du, “Mie-scattering calculation,” Appl. Opt. 43, 1951–1956 (2004).
    [CrossRef]
  24. L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).
  25. J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
    [CrossRef]
  26. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
  27. J. V. Dave, “Scattering of electromagnetic radiation by a large absorbing sphere,” IBM J. Res. Dev. 13, 302–313 (1969).
    [CrossRef]
  28. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef]
  29. C. Holloway, E. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
    [CrossRef]
  30. P. E. Falloon, “Theory and computation of spheroidal harmonics with general arguments,” Master of Science, The University of Western Australia (2001).
  31. C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).
    [CrossRef]

2013 (1)

Y. Li and N. Bowler, “Effects of parameter variations on negative effective constitutive parameters of nonmetallic metamaterials,” J. Appl. Phys. 113, 063501 (2013).
[CrossRef]

2012 (3)

Y. Li and N. Bowler, “Traveling waves on three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Trans. Antennas Propag. 60, 2727–2739 (2012).
[CrossRef]

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

R. Spurr, J. Wang, J. Zeng, and M. I. Mishchenko, “Linearized T-matrix and Mie scattering computations,” J. Quant. Spectrosc. Radiat. Transfer 113, 425–439 (2012).
[CrossRef]

2011 (2)

Y. Li and N. Bowler, “Analysis of double-negative (DNG) bandwidths for metamaterials composed of three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Antennas Wirel. Propag. Lett. 10, 1484–1487 (2011).
[CrossRef]

F. Xu and A. B. Davis, “Derivatives of light scattering properties of a nonspherical particle computed with the T-matrix method,” Opt. Lett. 36, 4464–4466 (2011).
[CrossRef]

2009 (2)

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

I. B. Vendik, O. G. Vendik, and M. A. Odit, “An isotropic metamaterial formed with ferroelectric ceramic spherical inclusions,” Phys. Solid State 51, 1590–1594 (2009).
[CrossRef]

2007 (2)

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).
[CrossRef]

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

2006 (1)

I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006).
[CrossRef]

2005 (2)

O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

G. Thomas, S. Bass, R. Grainger, and A. Lambert, “Retrieval of aerosol refractive index from extinction spectra with a damped harmonic-oscillator band model,” Appl. Opt. 44, 1332–1341 (2005).
[CrossRef]

2004 (2)

2003 (1)

C. Holloway, E. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[CrossRef]

1980 (1)

1976 (1)

1969 (2)

J. V. Dave, “Scattering of electromagnetic radiation by a large absorbing sphere,” IBM J. Res. Dev. 13, 302–313 (1969).
[CrossRef]

J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef]

1947 (1)

L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).

1908 (1)

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Baker-Jarvis, J.

C. Holloway, E. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[CrossRef]

Basilio, L. I.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Bass, S.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Bowler, N.

Y. Li and N. Bowler, “Effects of parameter variations on negative effective constitutive parameters of nonmetallic metamaterials,” J. Appl. Phys. 113, 063501 (2013).
[CrossRef]

Y. Li and N. Bowler, “Traveling waves on three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Trans. Antennas Propag. 60, 2727–2739 (2012).
[CrossRef]

Y. Li and N. Bowler, “Analysis of double-negative (DNG) bandwidths for metamaterials composed of three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Antennas Wirel. Propag. Lett. 10, 1484–1487 (2011).
[CrossRef]

Y. Li and N. Bowler, “Rational design of double-negative metamaterials consisting of 3D arrays of two different nonmetallic spheres arranged on a simple tetragonal lattice,” in 2011 IEEE International Symposium on Antennas and Propagation (2011), pp. 1494–1497.

Brener, I.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Clem, P. G.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Dave, J. V.

J. V. Dave, “Scattering of electromagnetic radiation by a large absorbing sphere,” IBM J. Res. Dev. 13, 302–313 (1969).
[CrossRef]

J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef]

Davis, A. B.

Du, H.

Ewen, G.

Falloon, P. E.

P. E. Falloon, “Theory and computation of spheroidal harmonics with general arguments,” Master of Science, The University of Western Australia (2001).

Ginn, J. C.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Grainger, R.

Hasekamp, O. P.

O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

Hines, P. F.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Holloway, C.

C. Holloway, E. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[CrossRef]

Holloway, C. L.

E. F. Kuester, K. L. Kumley, and C. L. Holloway, “Effect of sphere radius variation on the guided waves of a metafilm,” in 2013 IEEE International Symposium on Antennas and Propagation (2013).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Ihlefeld, J. F.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Infeld, L.

L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).

Kabos, P.

C. Holloway, E. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[CrossRef]

Kuester, E.

C. Holloway, E. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[CrossRef]

Kuester, E. F.

E. F. Kuester, K. L. Kumley, and C. L. Holloway, “Effect of sphere radius variation on the guided waves of a metafilm,” in 2013 IEEE International Symposium on Antennas and Propagation (2013).

K. L. Kumley and E. F. Kuester, “Effect of scatterer size variations on the reflection and transmission properties of a metafilm,” presented at National Radio Science Meeting, Boulder, CO (2012).

Kumley, K. L.

K. L. Kumley and E. F. Kuester, “Effect of scatterer size variations on the reflection and transmission properties of a metafilm,” presented at National Radio Science Meeting, Boulder, CO (2012).

E. F. Kuester, K. L. Kumley, and C. L. Holloway, “Effect of sphere radius variation on the guided waves of a metafilm,” in 2013 IEEE International Symposium on Antennas and Propagation (2013).

Lambert, A.

Landgraf, J.

O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

Li, Y.

Y. Li and N. Bowler, “Effects of parameter variations on negative effective constitutive parameters of nonmetallic metamaterials,” J. Appl. Phys. 113, 063501 (2013).
[CrossRef]

Y. Li and N. Bowler, “Traveling waves on three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Trans. Antennas Propag. 60, 2727–2739 (2012).
[CrossRef]

Y. Li and N. Bowler, “Analysis of double-negative (DNG) bandwidths for metamaterials composed of three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Antennas Wirel. Propag. Lett. 10, 1484–1487 (2011).
[CrossRef]

Y. Li and N. Bowler, “Rational design of double-negative metamaterials consisting of 3D arrays of two different nonmetallic spheres arranged on a simple tetragonal lattice,” in 2011 IEEE International Symposium on Antennas and Propagation (2011), pp. 1494–1497.

Lippens, D.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Lucas, J.

Mie, G.

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Mishchenko, M. I.

R. Spurr, J. Wang, J. Zeng, and M. I. Mishchenko, “Linearized T-matrix and Mie scattering computations,” J. Quant. Spectrosc. Radiat. Transfer 113, 425–439 (2012).
[CrossRef]

Odit, M.

I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006).
[CrossRef]

Odit, M. A.

I. B. Vendik, O. G. Vendik, and M. A. Odit, “An isotropic metamaterial formed with ferroelectric ceramic spherical inclusions,” Phys. Solid State 51, 1590–1594 (2009).
[CrossRef]

Peters, D. W.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Shore, R. A.

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

Simovski, C. R.

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).
[CrossRef]

Sinclair, M. B.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Spurr, R.

R. Spurr, J. Wang, J. Zeng, and M. I. Mishchenko, “Linearized T-matrix and Mie scattering computations,” J. Quant. Spectrosc. Radiat. Transfer 113, 425–439 (2012).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Stevens, J. O.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Stratton, J. A.

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

Thomas, G.

Tretyakov, S. A.

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).
[CrossRef]

van de Hulst, H. C.

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

Vendik, I.

I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006).
[CrossRef]

Vendik, I. B.

I. B. Vendik, O. G. Vendik, and M. A. Odit, “An isotropic metamaterial formed with ferroelectric ceramic spherical inclusions,” Phys. Solid State 51, 1590–1594 (2009).
[CrossRef]

Vendik, O.

I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006).
[CrossRef]

Vendik, O. G.

I. B. Vendik, O. G. Vendik, and M. A. Odit, “An isotropic metamaterial formed with ferroelectric ceramic spherical inclusions,” Phys. Solid State 51, 1590–1594 (2009).
[CrossRef]

Verner, B.

Wang, J.

R. Spurr, J. Wang, J. Zeng, and M. I. Mishchenko, “Linearized T-matrix and Mie scattering computations,” J. Quant. Spectrosc. Radiat. Transfer 113, 425–439 (2012).
[CrossRef]

Warne, L. K.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Wendt, J. R.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Wiscombe, W. J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Xu, F.

Yaghjian, A. D.

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

Zeng, J.

R. Spurr, J. Wang, J. Zeng, and M. I. Mishchenko, “Linearized T-matrix and Mie scattering computations,” J. Quant. Spectrosc. Radiat. Transfer 113, 425–439 (2012).
[CrossRef]

Zhang, F.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Zhao, Q.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Zhou, J.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Appl. Opt. (5)

IBM J. Res. Dev. (1)

J. V. Dave, “Scattering of electromagnetic radiation by a large absorbing sphere,” IBM J. Res. Dev. 13, 302–313 (1969).
[CrossRef]

IEEE Antennas Wirel. Propag. Lett. (1)

Y. Li and N. Bowler, “Analysis of double-negative (DNG) bandwidths for metamaterials composed of three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Antennas Wirel. Propag. Lett. 10, 1484–1487 (2011).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

Y. Li and N. Bowler, “Traveling waves on three-dimensional periodic arrays of two different magnetodielectric spheres arbitrarily arranged on a simple tetragonal lattice,” IEEE Trans. Antennas Propag. 60, 2727–2739 (2012).
[CrossRef]

C. Holloway, E. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[CrossRef]

J. Appl. Phys. (1)

Y. Li and N. Bowler, “Effects of parameter variations on negative effective constitutive parameters of nonmetallic metamaterials,” J. Appl. Phys. 113, 063501 (2013).
[CrossRef]

J. Geophys. Res. (1)

O. P. Hasekamp and J. Landgraf, “Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing,” J. Geophys. Res. 110, D04203 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

R. Spurr, J. Wang, J. Zeng, and M. I. Mishchenko, “Linearized T-matrix and Mie scattering computations,” J. Quant. Spectrosc. Radiat. Transfer 113, 425–439 (2012).
[CrossRef]

Mater. Today (1)

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

I. Vendik, O. Vendik, and M. Odit, “Isotropic artificial media with simultaneously negative permittivity and permeability,” Microw. Opt. Technol. Lett. 48, 2553–2556 (2006).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[CrossRef]

Phys. Solid State (1)

I. B. Vendik, O. G. Vendik, and M. A. Odit, “An isotropic metamaterial formed with ferroelectric ceramic spherical inclusions,” Phys. Solid State 51, 1590–1594 (2009).
[CrossRef]

Q. Appl. Math. (1)

L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).

Radio Sci. (1)

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

Other (9)

Y. Li and N. Bowler, “Rational design of double-negative metamaterials consisting of 3D arrays of two different nonmetallic spheres arranged on a simple tetragonal lattice,” in 2011 IEEE International Symposium on Antennas and Propagation (2011), pp. 1494–1497.

K. L. Kumley and E. F. Kuester, “Effect of scatterer size variations on the reflection and transmission properties of a metafilm,” presented at National Radio Science Meeting, Boulder, CO (2012).

E. F. Kuester, K. L. Kumley, and C. L. Holloway, “Effect of sphere radius variation on the guided waves of a metafilm,” in 2013 IEEE International Symposium on Antennas and Propagation (2013).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

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

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

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

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

P. E. Falloon, “Theory and computation of spheroidal harmonics with general arguments,” Master of Science, The University of Western Australia (2001).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Magnitudes of derivatives of a1 and b1 with respect to x [(a)–(b)], ϵr1 [(c)–(d)], and ϵr2 [(e)–(f)], obtained by presented formulas Eqs. (7)–(10), (13), (14), and (18)–(20), compared with those calculated by Mathematica. Results for |a1/μri|, |b1/μri| (i=1, 2) are not shown since they are similar to those shown for |a1/ϵri|, |b1/ϵri| (i=1, 2).

Fig. 2.
Fig. 2.

An array of identical spheres and unit cell geometry.

Fig. 3.
Fig. 3.

Ideal values and variation range of the effective relative permeability for an all-dielectric metamaterial consisting of a simple cubic array of identical tellurium nanoparticles, Fig. 2. Dotted line: values of μreff with Δm/m=4.2%; dashed line: ideal values of μreff; dashed–dotted line: values of μreff with Δm/m=5.9%; shaded area: variation range for Δm/m=14%. Inset shows the wavelength region, in which the min(μreff,idl+Δμreff) becomes equal to zero.

Tables (1)

Tables Icon

Table 1. Accuracy and CPU Time for Computation of Mie Derivatives by Analytical Expressions, Approximate Expressions, and by Numerical Differentiation for Scattering by a Single Magnetodielectric Sphere with Parameters: ϵr1=40, μr1=200, ϵr2=μr2=1, and x=0.5 [29]. Results for bn/x, bn/ϵri and bn/μri (i=1, 2) are not Shown Since they are Similar to Those Shown for an/x, an/ϵri, and an/μri (i=1, 2)

Equations (51)

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

an=μr1ψn(m1x)ψn(m2x)μr2mψn(m1x)ψn(m2x)μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x),
bn=μr1ψn(m1x)ψn(m2x)μr2mψn(m1x)ψn(m2x)μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x),
ψn(z)zjn(z)andξn(z)zhn(1)(z),
ψn(z)ξn(z)ψn(z)ξn(z)=i,
ψn(z)ξn(z)ψn(z)ξn(z)=0,
ψn(z)ξn(z)ψn(z)ξn(z)=i[1n(n+1)z2],
anx=i{μr2(μr2μr1)m12m2[ψn(m1x)]2+μr1μr2m12m2ψn(m1x)ψn(m1x)+μr12m2[ψn(m1x)]2[1n(n+1)(m2x)2][μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
bnx=i{μr1(μr1m2μr2m12m2)[ψn(m1x)]2+μr1μr2m12m2ψn(m1x)ψn(m1x)+μr22m12m2[ψn(m1x)]2[1n(n+1)(m2x)2][μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
anϵr1=0.5i{μr12μr2ϵr2x{ψn(m1x)ψn(m1x)[ψn(m1x)]2}+μr11.5μr2ϵr1ϵr2ψn(m1x)ψn(m1x)[μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
bnϵr1=0.5i{μr12μr2ϵr2x{ψn(m1x)ψn(m1x)[ψn(m1x)]2}μr11.5μr2ϵr1ϵr2ψn(m1x)ψn(m1x)[μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
anμr1=0.5i{ϵr1μr1μr2ϵr2x{ψn(m1x)ψn(m1x)[ψn(m1x)]2}ϵr1μr1μr2ϵr2ψn(m1x)ψn(m1x)[μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
bnμr1=0.5i{ϵr1μr1μr2ϵr2x{ψn(m1x)ψn(m1x)[ψn(m1x)]2}+ϵr1μr1μr2ϵr2ψn(m1x)ψn(m1x)[μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
anϵr2=0.5i{ϵr1μr1(μr2ϵr2)1.5x[ψn(m1x)]2+μr12μr2ϵr2x[ψn(m1x)]2[1n(n+1)(m2x)2]ϵr1μr2(μr1ϵr2)1.5ψn(m1x)ψn(m1x)[μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
bnϵr2=0.5i{μr12μr2ϵr2x[ψn(m1x)]2+ϵr1μr1(μr2ϵr2)1.5x[ψn(m1x)]2[1n(n+1)(m2x)2]+ϵr1μr2(μr1ϵr2)1.5ψn(m1x)ψn(m1x)[μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
anμr2=0.5i{ϵr1μr1μr2ϵr2x[ψn(m1x)]2+μr12ϵr2μr2x[ψn(m1x)]2[1n(n+1)(m2x)2]+μr11.5ϵr1ϵr2μr2ψn(m1x)ψn(m1x)[μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2},
bnμr2=0.5i{μr12ϵr2μr2x[ψn(m1x)]2+ϵr1μr1μr2ϵr2x[ψn(m1x)]2[1n(n+1)(m2x)2]μr11.5ϵr1ϵr2μr2ψn(m1x)ψn(m1x)[μr1ψn(m1x)ξn(m2x)μr2mψn(m1x)ξn(m2x)]2}.
z2w(z)+[z2n(n+1)]w(z)=0,
ψn(z)=ψn(z)[n(n+1)z21].
ψn(z)=ψn1(z)nzψn(z).
ψn(z)=2n1zψn1(z)ψn2(z),
An(z)=ψn(z)ψn(z),
An(z)=n+1z1An+1(z)+n+1z,
anx=i{μr2(μr2μr1)m12m2[An(m1x)]2+μr1μr2m12m2ψn(m1x)ψn(m1x)+μr12m2[1n(n+1)(m2x)2][μr1ξn(m2x)μr2mAn(m1x)ξn(m2x)]2}.
anx=i{μr2(μr2μr1)m12m2[An(m1x)]2μr1μr2m12m2[1n(n+1)(m1x)2]+μr12m2[1n(n+1)(m2x)2][μr1ξn(m2x)μr2mAn(m1x)ξn(m2x)]2}.
anx=i{μr2(μr2μr1)m12m2[An(m1x)]2μr1μr2m12m2[1n(n+1)(m1x)2]+μr12m2[1n(n+1)(m2x)2]{[μr2mAn(m1x)+μr1nm2x]ξn(m2x)μr1ξn1(m2x)}2}.
bnx=i{μr1(μr1m2μr2m12m2)[An(m1x)]2μr1μr2m12m2[1n(n+1)(m1x)2]+μr22m12m2[1n(n+1)(m2x)2]{[μr1An(m1x)+μr2mnm2x]ξn(m2x)μr2mξn1(m2x)}2},
anϵr1=0.5i{μr12μr2ϵr2x[n(n+1)(m1x)21[An(m1x)]2]+μr11.5μr2ϵr1ϵr2An(m1x){[μr2mAn(m1x)+μr1nm2x]ξn(m2x)μr1ξn1(m2x)}2},
bnϵr1=0.5i{μr12μr2ϵr2x[n(n+1)(m1x)21[An(m1x)]2]μr11.5μr2ϵr1ϵr2An(m1x){[μr1An(m1x)+μr2mnm2x]ξn(m2x)μr2mξn1(m2x)}2},
anμr1=0.5i{ϵr1μr1μr2ϵr2x[n(n+1)(m1x)21[An(m1x)]2]ϵr1μr1μr2ϵr2An(m1x){[μr2mAn(m1x)+μr1nm2x]ξn(m2x)μr1ξn1(m2x)}2},
bnμr1=0.5i{ϵr1μr1μr2ϵr2x[n(n+1)(m1x)21[An(m1x)]2]+ϵr1μr1μr2ϵr2An(m1x){[μr1An(m1x)+μr2mnm2x]ξn(m2x)μr2mξn1(m2x)}2},
anϵr2=0.5i{ϵr1μr1(μr2ϵr2)1.5x[An(m1x)]2ϵr1μr2(μr1ϵr2)1.5An(m1x)+μr12μr2ϵr2x[1n(n+1)(m1x)2]{[μr2mAn(m1x)+μr1nm2x]ξn(m2x)μr1ξn1(m2x)}2},
bnϵr2=0.5i{μr12μr2ϵr2x[An(m1x)]2+ϵr1μr2(μr1ϵr2)1.5An(m1x)+ϵr1μr1(μr2ϵr2)1.5x[1n(n+1)(m1x)2]{[μr1An(m1x)+μr2mnm2x]ξn(m2x)μr2mξn1(m2x)}2},
anμr2=0.5i{ϵr1μr1μr2ϵr2x[An(m1x)]2+μr11.5ϵr1ϵr2μr2An(m1x)+μr12ϵr2μr2x[1n(n+1)(m2x)2]{[μr2mAn(m1x)+μr1nm2x]ξn(m2x)μr1ξn1(m2x)}2},
bnμr2=0.5i{μr12ϵr2μr2x[An(m1x)]2μr11.5ϵr1ϵr2μr2An(m1x)+ϵr1μr1μr2ϵr2x[1n(n+1)(m2x)2]{[μr1An(m1x)+μr2mnm2x]ξn(m2x)μr2mξn1(m2x)}2}.
fx=f(x+h)f(xh)2h,
ϵreff=ϵr22B1+33B1
B1=6πib1(k0d)3(ϵr2μr2)1.5,
Δϵreff=mϵreffmΔm,
ϵreffm=9ϵr2(3B1)2B1m
B1m=6πi(k0d)3(ϵr2μr2)1.5b1m
B1μr2=6πi(k0d)3(ϵr2μr2)1.5[b1μr21.5(μr2)1b1],
B1(k0d)=18πib1(k0d)4(ϵr2μr2)1.5.
ϵreffϵr2=2B1+33B1+9ϵr2(3B1)2B1ϵr2
B1ϵr2=6πi(k0d)3(ϵr2μr2)1.5[b1ϵr21.5(ϵr2)1b1].
μreffm=9μr2(3B1)2B1m
B1m=6πi(k0d)3(ϵr2μr2)1.5a1m
B1ϵr2=6πi(k0d)3(ϵr2μr2)1.5[a1ϵr21.5(ϵr2)1a1],
B1(k0d)=18πia1(k0d)4(ϵr2μr2)1.5.
μreffμr2=2B1+33B1+9μr2(3B1)2B1μr2
B1μr2=6πi(k0d)3(ϵr2μr2)1.5[a1μr21.5(μr2)1a1].
μreff,idlΔμreff<μreff<μreff,idl+Δμreff,

Metrics