Abstract

Expressions of scattering coefficients for calculating scattering by large chiral spheres are derived by using logarithmic derivatives and ratios of Riccati–Bessel functions. The improved expressions can be easily applied to the case of an arbitrarily shaped beam incidence. A simplified expression of the scattered field in the far field is obtained for the case of x-polarized plane-wave incidence. To verify the correctness and accuracy of the theory and codes, our results are compared with those in literature and those calculated by Mie theory. Radar cross sections of a large chiral sphere are numerically studied. It is found that the rainbow phenomenon of a chiral sphere is very different from that of an isotropic sphere.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer, 1989).
  2. D. J. Gordon, “Mie scattering by optically active particles,” Biochemistry 11, 413–420 (1972).
    [CrossRef]
  3. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
    [CrossRef]
  4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
  5. M. Hinders and B. Rhodes, “Electromagnetic-wave scattering from chiral spheres in chiral media,” Nuovo Cimento D 14, 575–583 (1992).
    [CrossRef]
  6. M. F. R. Cooray and I. R. Ciric, “Wave scattering by a chiral spheroid,” J. Opt. Soc. Am. A 10, 1197–1203 (1993).
    [CrossRef]
  7. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Scattering and absorption characteristics of lossy dielectric, chiral, nonspherical objects,” Appl. Opt. 24, 4146–4154 (1985).
    [CrossRef]
  8. D. L. Jaggard and J. C. Liu, “The matrix Riccati equation for scattering from stratified chiral spheres,” IEEE Trans. Antennas Propag. 47, 1201–1207 (1999).
    [CrossRef]
  9. L. Li, Y. Dan, M. Leong, and J. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” Prog. Electromagn. Res. 13, 1203–1206 (1999).
    [CrossRef]
  10. V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Antennas Propag. Mag. 46(5), 94–99 (2004).
    [CrossRef]
  11. D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. 41, 526–533 (2011).
    [CrossRef]
  12. D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
    [CrossRef]
  13. M. Yuceer, J. R. Mautz, and E. Arvas, “Method of moments solution for the radar cross section of a chiral body of revolution,” IEEE Trans. Antennas Propag. 53, 1163–1167 (2005).
    [CrossRef]
  14. M. Hasanovic, M. Chong, J. R. Mautz, and E. Arvas, “Scattering from 3-D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag. 55, 1817–1825 (2007).
    [CrossRef]
  15. C. Mei, M. Hasanovic, J. K. Lee, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three dimensional inhomogeneous bianisotropic body,” PIERS Online 3, 680–684 (2007).
    [CrossRef]
  16. V. Demir, A. Z. Elsherbeni, and E. Arvas, “FDTD formulations for scattering from three dimensional chiral objects,” in The 20th Annual Review of Progress in Applied Computational Electromagnetics Society (ACES, 2004), record 577.
  17. L. Kuzu, V. Demir, A. Z. Elsherbeni, and E. Arvas, “Electromagnetic scattering from arbitrarily shaped chiral objects using the finite difference frequency domain method,” Progress Electromagn. Res. 67, 1–24 (2007).
    [CrossRef]
  18. A. L. Aden, and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  19. A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994), Vol. 2.
  20. D. Sarkar, and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102 (1997).
    [CrossRef]
  21. L. Infeld, “The influence of the width of the gap upon the theory of antennas,” Q. Appl. Math. 5, 113–132 (1947).
  22. A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
    [CrossRef]
  23. J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Devel. 13, 302–313 (1969).
    [CrossRef]
  24. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 15, 668–671 (1976).
    [CrossRef]
  25. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef]
  26. G. W. Kattawar, and G. N. Plass, “Electromagnetic scattering from absorbing spheres,” Appl. Opt. 6, 1377–1382 (1967).
    [CrossRef]
  27. O. B. Toon, and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
    [CrossRef]
  28. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef]
  29. Z. S. Wu, and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
    [CrossRef]
  30. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
    [CrossRef]
  31. F. Xu, K. Ren, and X. Cai, “Expansion of an arbitrarily oriented, located, and shaped beam in spheroidal coordinates,” J. Opt. Soc. Am. A 24, 109–118 (2007).
    [CrossRef]
  32. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  33. G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef]
  34. A. Doicu, and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz–Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
    [CrossRef]
  35. J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
    [CrossRef]
  36. Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572–576 (2009).
    [CrossRef]
  37. J. Munoz, M. Rojo, A. Parrefio, and J. Margineda, “Automatic measurement of permittivity and permeability at microwave frequencies using normal and oblique free-wave incidence with focused beam,” IEEE Trans. Instrum. Meas. 47, 886–892 (1998).
    [CrossRef]
  38. A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
    [CrossRef]
  39. V. D. Hulst, Light Scattering by Small Particles (Wiley, 1957).

2011

D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. 41, 526–533 (2011).
[CrossRef]

2009

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572–576 (2009).
[CrossRef]

2008

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

2007

M. Hasanovic, M. Chong, J. R. Mautz, and E. Arvas, “Scattering from 3-D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag. 55, 1817–1825 (2007).
[CrossRef]

C. Mei, M. Hasanovic, J. K. Lee, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three dimensional inhomogeneous bianisotropic body,” PIERS Online 3, 680–684 (2007).
[CrossRef]

L. Kuzu, V. Demir, A. Z. Elsherbeni, and E. Arvas, “Electromagnetic scattering from arbitrarily shaped chiral objects using the finite difference frequency domain method,” Progress Electromagn. Res. 67, 1–24 (2007).
[CrossRef]

F. Xu, K. Ren, and X. Cai, “Expansion of an arbitrarily oriented, located, and shaped beam in spheroidal coordinates,” J. Opt. Soc. Am. A 24, 109–118 (2007).
[CrossRef]

2005

M. Yuceer, J. R. Mautz, and E. Arvas, “Method of moments solution for the radar cross section of a chiral body of revolution,” IEEE Trans. Antennas Propag. 53, 1163–1167 (2005).
[CrossRef]

2004

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Antennas Propag. Mag. 46(5), 94–99 (2004).
[CrossRef]

2003

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

1999

D. L. Jaggard and J. C. Liu, “The matrix Riccati equation for scattering from stratified chiral spheres,” IEEE Trans. Antennas Propag. 47, 1201–1207 (1999).
[CrossRef]

L. Li, Y. Dan, M. Leong, and J. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” Prog. Electromagn. Res. 13, 1203–1206 (1999).
[CrossRef]

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
[CrossRef]

1998

J. Munoz, M. Rojo, A. Parrefio, and J. Margineda, “Automatic measurement of permittivity and permeability at microwave frequencies using normal and oblique free-wave incidence with focused beam,” IEEE Trans. Instrum. Meas. 47, 886–892 (1998).
[CrossRef]

1997

1995

1993

1992

M. Hinders and B. Rhodes, “Electromagnetic-wave scattering from chiral spheres in chiral media,” Nuovo Cimento D 14, 575–583 (1992).
[CrossRef]

1991

Z. S. Wu, and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1988

1985

1981

1980

1976

1974

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

1972

D. J. Gordon, “Mie scattering by optically active particles,” Biochemistry 11, 413–420 (1972).
[CrossRef]

1969

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

1967

1951

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
[CrossRef]

A. L. Aden, and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1947

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

Ackerman, T. P.

Aden, A. L.

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
[CrossRef]

A. L. Aden, and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Arvas, E.

C. Mei, M. Hasanovic, J. K. Lee, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three dimensional inhomogeneous bianisotropic body,” PIERS Online 3, 680–684 (2007).
[CrossRef]

M. Hasanovic, M. Chong, J. R. Mautz, and E. Arvas, “Scattering from 3-D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag. 55, 1817–1825 (2007).
[CrossRef]

L. Kuzu, V. Demir, A. Z. Elsherbeni, and E. Arvas, “Electromagnetic scattering from arbitrarily shaped chiral objects using the finite difference frequency domain method,” Progress Electromagn. Res. 67, 1–24 (2007).
[CrossRef]

M. Yuceer, J. R. Mautz, and E. Arvas, “Method of moments solution for the radar cross section of a chiral body of revolution,” IEEE Trans. Antennas Propag. 53, 1163–1167 (2005).
[CrossRef]

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Antennas Propag. Mag. 46(5), 94–99 (2004).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

V. Demir, A. Z. Elsherbeni, and E. Arvas, “FDTD formulations for scattering from three dimensional chiral objects,” in The 20th Annual Review of Progress in Applied Computational Electromagnetics Society (ACES, 2004), record 577.

Bhandari, R.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Bohren, F.

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

Cai, X.

Chong, M.

M. Hasanovic, M. Chong, J. R. Mautz, and E. Arvas, “Scattering from 3-D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag. 55, 1817–1825 (2007).
[CrossRef]

Ciric, I. R.

Cooray, M. F. R.

Dan, Y.

L. Li, Y. Dan, M. Leong, and J. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” Prog. Electromagn. Res. 13, 1203–1206 (1999).
[CrossRef]

Dave, J. V.

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

Demir, V.

L. Kuzu, V. Demir, A. Z. Elsherbeni, and E. Arvas, “Electromagnetic scattering from arbitrarily shaped chiral objects using the finite difference frequency domain method,” Progress Electromagn. Res. 67, 1–24 (2007).
[CrossRef]

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Antennas Propag. Mag. 46(5), 94–99 (2004).
[CrossRef]

V. Demir, A. Z. Elsherbeni, and E. Arvas, “FDTD formulations for scattering from three dimensional chiral objects,” in The 20th Annual Review of Progress in Applied Computational Electromagnetics Society (ACES, 2004), record 577.

Doicu, A.

Elsherbeni, A.

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Antennas Propag. Mag. 46(5), 94–99 (2004).
[CrossRef]

Elsherbeni, A. Z.

L. Kuzu, V. Demir, A. Z. Elsherbeni, and E. Arvas, “Electromagnetic scattering from arbitrarily shaped chiral objects using the finite difference frequency domain method,” Progress Electromagn. Res. 67, 1–24 (2007).
[CrossRef]

V. Demir, A. Z. Elsherbeni, and E. Arvas, “FDTD formulations for scattering from three dimensional chiral objects,” in The 20th Annual Review of Progress in Applied Computational Electromagnetics Society (ACES, 2004), record 577.

Geng, Y. L.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572–576 (2009).
[CrossRef]

Gomez, A.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

Gordon, D. J.

D. J. Gordon, “Mie scattering by optically active particles,” Biochemistry 11, 413–420 (1972).
[CrossRef]

Gouesbet, G.

Grehan, G.

Gréhan, G.

Guzatov, D.

D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. 41, 526–533 (2011).
[CrossRef]

Halas, N. J.

D. Sarkar, and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102 (1997).
[CrossRef]

Hasanovic, M.

M. Hasanovic, M. Chong, J. R. Mautz, and E. Arvas, “Scattering from 3-D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag. 55, 1817–1825 (2007).
[CrossRef]

C. Mei, M. Hasanovic, J. K. Lee, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three dimensional inhomogeneous bianisotropic body,” PIERS Online 3, 680–684 (2007).
[CrossRef]

Hinders, M.

M. Hinders and B. Rhodes, “Electromagnetic-wave scattering from chiral spheres in chiral media,” Nuovo Cimento D 14, 575–583 (1992).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Hulst, V. D.

V. D. Hulst, Light Scattering by Small Particles (Wiley, 1957).

Infeld, L.

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

Jaggard, D. L.

D. L. Jaggard and J. C. Liu, “The matrix Riccati equation for scattering from stratified chiral spheres,” IEEE Trans. Antennas Propag. 47, 1201–1207 (1999).
[CrossRef]

Kattawar, G. W.

Kerker, M.

A. L. Aden, and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Klimov, V. V.

D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. 41, 526–533 (2011).
[CrossRef]

Kong, J.

L. Li, Y. Dan, M. Leong, and J. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” Prog. Electromagn. Res. 13, 1203–1206 (1999).
[CrossRef]

Kuzu, L.

L. Kuzu, V. Demir, A. Z. Elsherbeni, and E. Arvas, “Electromagnetic scattering from arbitrarily shaped chiral objects using the finite difference frequency domain method,” Progress Electromagn. Res. 67, 1–24 (2007).
[CrossRef]

Lakhtakia, A.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Scattering and absorption characteristics of lossy dielectric, chiral, nonspherical objects,” Appl. Opt. 24, 4146–4154 (1985).
[CrossRef]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer, 1989).

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994), Vol. 2.

Lee, J. K.

C. Mei, M. Hasanovic, J. K. Lee, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three dimensional inhomogeneous bianisotropic body,” PIERS Online 3, 680–684 (2007).
[CrossRef]

Lentz, W. J.

Leong, M.

L. Li, Y. Dan, M. Leong, and J. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” Prog. Electromagn. Res. 13, 1203–1206 (1999).
[CrossRef]

Li, L.

L. Li, Y. Dan, M. Leong, and J. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” Prog. Electromagn. Res. 13, 1203–1206 (1999).
[CrossRef]

Liu, J. C.

D. L. Jaggard and J. C. Liu, “The matrix Riccati equation for scattering from stratified chiral spheres,” IEEE Trans. Antennas Propag. 47, 1201–1207 (1999).
[CrossRef]

Lock, J. A.

Maheu, B.

Margineda, J.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

J. Munoz, M. Rojo, A. Parrefio, and J. Margineda, “Automatic measurement of permittivity and permeability at microwave frequencies using normal and oblique free-wave incidence with focused beam,” IEEE Trans. Instrum. Meas. 47, 886–892 (1998).
[CrossRef]

Mautz, J. R.

M. Hasanovic, M. Chong, J. R. Mautz, and E. Arvas, “Scattering from 3-D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag. 55, 1817–1825 (2007).
[CrossRef]

M. Yuceer, J. R. Mautz, and E. Arvas, “Method of moments solution for the radar cross section of a chiral body of revolution,” IEEE Trans. Antennas Propag. 53, 1163–1167 (2005).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

Mei, C.

C. Mei, M. Hasanovic, J. K. Lee, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three dimensional inhomogeneous bianisotropic body,” PIERS Online 3, 680–684 (2007).
[CrossRef]

Molina-Cuberos, G. J.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

Munoz, J.

J. Munoz, M. Rojo, A. Parrefio, and J. Margineda, “Automatic measurement of permittivity and permeability at microwave frequencies using normal and oblique free-wave incidence with focused beam,” IEEE Trans. Instrum. Meas. 47, 886–892 (1998).
[CrossRef]

Nuez, M. J.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

Parrefio, A.

J. Munoz, M. Rojo, A. Parrefio, and J. Margineda, “Automatic measurement of permittivity and permeability at microwave frequencies using normal and oblique free-wave incidence with focused beam,” IEEE Trans. Instrum. Meas. 47, 886–892 (1998).
[CrossRef]

Plass, G. N.

Qiu, C. W.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572–576 (2009).
[CrossRef]

Ren, K.

Rhodes, B.

M. Hinders and B. Rhodes, “Electromagnetic-wave scattering from chiral spheres in chiral media,” Nuovo Cimento D 14, 575–583 (1992).
[CrossRef]

Rojo, M.

J. Munoz, M. Rojo, A. Parrefio, and J. Margineda, “Automatic measurement of permittivity and permeability at microwave frequencies using normal and oblique free-wave incidence with focused beam,” IEEE Trans. Instrum. Meas. 47, 886–892 (1998).
[CrossRef]

Saiz Ipina, J. A.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

Sarkar, D.

D. Sarkar, and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102 (1997).
[CrossRef]

Solano, M. A.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

Toon, O. B.

Varadan, V. K.

Varadan, V. V.

Vegas, A.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

Wang, Y. P.

Z. S. Wu, and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wiscombe, W. J.

Worasawate, D.

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Antennas Propag. Mag. 46(5), 94–99 (2004).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

Wriedt, T.

Wu, Z. S.

Z. S. Wu, and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Xu, F.

Yuan, N.

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572–576 (2009).
[CrossRef]

Yuceer, M.

M. Yuceer, J. R. Mautz, and E. Arvas, “Method of moments solution for the radar cross section of a chiral body of revolution,” IEEE Trans. Antennas Propag. 53, 1163–1167 (2005).
[CrossRef]

Appl. Opt.

Biochemistry

D. J. Gordon, “Mie scattering by optically active particles,” Biochemistry 11, 413–420 (1972).
[CrossRef]

Chem. Phys. Lett.

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

IBM J. Res. Devel.

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

IEEE Antennas Propag. Mag.

V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Antennas Propag. Mag. 46(5), 94–99 (2004).
[CrossRef]

IEEE Trans. Antennas Propag.

D. L. Jaggard and J. C. Liu, “The matrix Riccati equation for scattering from stratified chiral spheres,” IEEE Trans. Antennas Propag. 47, 1201–1207 (1999).
[CrossRef]

D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag. 51, 1077–1084 (2003).
[CrossRef]

M. Yuceer, J. R. Mautz, and E. Arvas, “Method of moments solution for the radar cross section of a chiral body of revolution,” IEEE Trans. Antennas Propag. 53, 1163–1167 (2005).
[CrossRef]

M. Hasanovic, M. Chong, J. R. Mautz, and E. Arvas, “Scattering from 3-D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag. 55, 1817–1825 (2007).
[CrossRef]

Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antennas Propag. 57, 572–576 (2009).
[CrossRef]

IEEE Trans. Instrum. Meas.

J. Munoz, M. Rojo, A. Parrefio, and J. Margineda, “Automatic measurement of permittivity and permeability at microwave frequencies using normal and oblique free-wave incidence with focused beam,” IEEE Trans. Instrum. Meas. 47, 886–892 (1998).
[CrossRef]

IEEE Trans. Microwave Theor. Tech.

A. Gomez, A. Lakhtakia, J. Margineda, G. J. Molina-Cuberos, M. J. Nuez, J. A. Saiz Ipina, A. Vegas, and M. A. Solano, “Full-wave hybrid technique for 3-D isotropic-chiral-material discontinuities in rectangular waveguides: theory and experiment,” IEEE Trans. Microwave Theor. Tech. 56, 2815–2825 (2008).
[CrossRef]

J. Appl. Phys.

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
[CrossRef]

A. L. Aden, and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Opt. Soc. Am. A

Nuovo Cimento D

M. Hinders and B. Rhodes, “Electromagnetic-wave scattering from chiral spheres in chiral media,” Nuovo Cimento D 14, 575–583 (1992).
[CrossRef]

Phys. Rev. E

D. Sarkar, and N. J. Halas, “General vector basis function solution of Maxwell’s equations,” Phys. Rev. E 56, 1102 (1997).
[CrossRef]

PIERS Online

C. Mei, M. Hasanovic, J. K. Lee, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three dimensional inhomogeneous bianisotropic body,” PIERS Online 3, 680–684 (2007).
[CrossRef]

Prog. Electromagn. Res.

L. Li, Y. Dan, M. Leong, and J. Kong, “Electromagnetic scattering by an inhomogeneous chiral sphere of varying permittivity: a discrete analysis using multilayered model,” Prog. Electromagn. Res. 13, 1203–1206 (1999).
[CrossRef]

Progress Electromagn. Res.

L. Kuzu, V. Demir, A. Z. Elsherbeni, and E. Arvas, “Electromagnetic scattering from arbitrarily shaped chiral objects using the finite difference frequency domain method,” Progress Electromagn. Res. 67, 1–24 (2007).
[CrossRef]

Q. Appl. Math.

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

Quantum Electron.

D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. 41, 526–533 (2011).
[CrossRef]

Radio Sci.

Z. S. Wu, and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Other

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

V. D. Hulst, Light Scattering by Small Particles (Wiley, 1957).

V. Demir, A. Z. Elsherbeni, and E. Arvas, “FDTD formulations for scattering from three dimensional chiral objects,” in The 20th Annual Review of Progress in Applied Computational Electromagnetics Society (ACES, 2004), record 577.

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994), Vol. 2.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer, 1989).

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 (7)

Fig. 1.
Fig. 1.

Bistatic RCS compared with those in reference in E plane. λ = 0.3 m , a = 0.072 m , ε r = 4 , μ r = 1 , and κ = 0 , 0.5. (a) Co-polarized σ θ ; (b) cross-polarized σ ϕ .

Fig. 2.
Fig. 2.

Comparison with Mie Theory. a = 100 λ , ε r = 1.7689 , μ r = 1.0 , and κ = 0 .

Fig. 3.
Fig. 3.

RCS σ θ and σ ϕ of a large chiral sphere. a = 50 λ , ε r = 4.0 , μ r = 1.0 , and κ = 0.01 . (a)  E plane; (b)  H plane.

Fig. 4.
Fig. 4.

Comparison between σ ϕ in E plane and σ θ in H plane. The parameters are the sameas in Fig. 3.

Fig. 5.
Fig. 5.

Effects of the chirality parameter on RCS. a = 100 λ , ε r = 1.7688 + 0.0266 i , μ r = 1.0 , and κ = 0.0 , 0.1, 0.3, 0.5. (a)  E plane; (b)  H plane.

Fig. 6.
Fig. 6.

RCS of a large chiral sphere with complex parameters. a = 100 λ , ε r = 1.7688 + 0.0266 i , μ r = 0.85 + 0.02 i , and κ = 0.1 , 0.1 + 0.015 i , and 0.1 + 0.0236 i .

Fig. 7.
Fig. 7.

Rainbow phenomenon of a large chiral sphere. a = 500 λ , ε r = 1.7689 , μ r = 1.0 , and κ = 0.08 . (a)  E plane; (b)  H plane.

Equations (38)

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

D = ε c E + i κ ε 0 μ 0 H , B = i κ ε 0 μ 0 E + μ c H ,
E i = E 0 n = 1 m = n n [ a m n i M m n ( 1 ) ( r , k ) + b m n i N m n ( 1 ) ( r , k ) ] ,
H i = k E 0 i ω μ n = 1 m = n n [ a m n i N m n ( 1 ) ( r , k ) + b m n i M m n ( 1 ) ( r , k ) ] ,
E s = E 0 n = 1 m = n n [ A m n s M m n ( 3 ) ( r , k ) + B m n s N m n ( 3 ) ( r , k ) ] ,
H s = k E 0 i ω μ n = 1 m = n n [ A m n s N m n ( 3 ) ( r , k ) + B m n s M m n ( 3 ) ( r , k ) ] ,
E = n = 1 m = n n [ A m n M m n ( 1 ) ( r , k 1 ) + A m n N m n ( 1 ) ( r , k 1 ) + B m n M m n ( 1 ) ( r , k 2 ) B m n N m n ( 1 ) ( r , k 2 ) ] ,
H = Q n = 1 m = n n [ A m n N m n ( 1 ) ( r , k 1 ) + A m n M m n ( 1 ) ( r , k 1 ) + B m n N m n ( 1 ) ( r , k 2 ) B m n M m n ( 1 ) ( r , k 2 ) ] ,
A m n s = D G B F A D B C , B m n s = A F C G A D B C .
A = ξ n ( x 0 ) ψ n ( x 1 ) η r ξ n ( x 0 ) ψ n ( x 1 ) ,
B = η r ξ n ( x 0 ) ψ n ( x 1 ) ξ n ( x 0 ) ψ n ( x 1 ) ,
G = a m n i [ η r ψ n ( x 0 ) ψ n ( x 1 ) ψ n ( x 0 ) ψ n ( x 1 ) ] + b m n i [ ψ n ( x 0 ) ψ n ( x 1 ) η r ψ n ( x 0 ) ψ n ( x 1 ) ] ,
C = ξ n ( x 0 ) ψ n ( x 2 ) η r ψ n ( x 2 ) ξ n ( x 0 ) ,
D = ψ n ( x 2 ) ξ n ( x 0 ) η r ξ n ( x 0 ) ψ n ( x 2 ) ,
F = a m n i [ η r ψ n ( x 2 ) ψ n ( x 0 ) ψ n ( x 0 ) ψ n ( x 2 ) ] + b m n i [ η r ψ n ( x 0 ) ψ n ( x 2 ) ψ n ( x 2 ) ψ n ( x 0 ) ] ,
A m n s = A n s a a m n i + A n s b b m n i , B m n s = B n s a a m n i + B n s b b m n i ,
A n s a = ψ n ( x 0 ) ξ n ( x 0 ) D n ( 1 ) ( x 1 ) η r D n ( 1 ) ( x 0 ) η r D n ( 1 ) ( x 1 ) D n ( 3 ) ( x 0 ) + D n ( 1 ) ( x 2 ) η r D n ( 1 ) ( x 0 ) η r D n ( 1 ) ( x 2 ) D n ( 3 ) ( x 0 ) η r D n ( 3 ) ( x 0 ) D n ( 1 ) ( x 1 ) η r D n ( 1 ) ( x 1 ) D n ( 3 ) ( x 0 ) + η r D n ( 3 ) ( x 0 ) D n ( 1 ) ( x 2 ) η r D n ( 1 ) ( x 2 ) D n ( 3 ) ( x 0 ) ,
A n s b = ψ n ( x 0 ) ξ n ( x 0 ) η r D n ( 1 ) ( x 1 ) D n ( 1 ) ( x 0 ) η r D n ( 1 ) ( x 1 ) D n ( 3 ) ( x 0 ) η r D n ( 1 ) ( x 2 ) D n ( 1 ) ( x 0 ) η r D n ( 1 ) ( x 2 ) D n ( 3 ) ( x 0 ) η r D n ( 3 ) ( x 0 ) D n ( 1 ) ( x 1 ) η r D n ( 1 ) ( x 1 ) D n ( 3 ) ( x 0 ) + η r D n ( 3 ) ( x 0 ) D n ( 1 ) ( x 2 ) η r D n ( 1 ) ( x 2 ) D n ( 3 ) ( x 0 ) ,
B n s a = A n s b ,
B n s b = ψ n ( x 0 ) ξ n ( x 0 ) η r D n ( 1 ) ( x 1 ) D n ( 1 ) ( x 0 ) D n ( 1 ) ( x 1 ) η r D n ( 3 ) ( x 0 ) + η r D n ( 1 ) ( x 2 ) D n ( 1 ) ( x 0 ) D n ( 1 ) ( x 2 ) η r D n ( 3 ) ( x 0 ) D n ( 3 ) ( x 0 ) η r D n ( 1 ) ( x 1 ) D n ( 1 ) ( x 1 ) η r D n ( 3 ) ( x 0 ) + D n ( 3 ) ( x 0 ) η r D n ( 1 ) ( x 2 ) D n ( 1 ) ( x 2 ) η r D n ( 3 ) ( x 0 ) .
A m n = E 0 η r k 1 k 0 ψ n ( x 0 ) ψ n ( x 1 ) D n ( 1 ) ( x 0 ) D n ( 3 ) ( x 0 ) D n ( 1 ) ( x 2 ) η r D n ( 3 ) ( x 0 ) a m n i + D n ( 1 ) ( x 0 ) D n ( 3 ) ( x 0 ) η r D n ( 1 ) ( x 2 ) D n ( 3 ) ( x 0 ) b m n i D n ( 1 ) ( x 1 ) η r D n ( 3 ) ( x 0 ) D n ( 1 ) ( x 2 ) η r D n ( 3 ) ( x 0 ) + η r D n ( 1 ) ( x 1 ) D n ( 3 ) ( x 0 ) η r D n ( 1 ) ( x 2 ) D n ( 3 ) ( x 0 ) ,
B m n = E 0 η r k 2 k 0 ψ n ( x 0 ) ψ n ( x 2 ) D n ( 1 ) ( x 0 ) D n ( 3 ) ( x 0 ) D n ( 1 ) ( x 1 ) η r D n ( 3 ) ( x 0 ) a m n i D n ( 1 ) ( x 0 ) D n ( 3 ) ( x 0 ) η r D n ( 1 ) ( x 1 ) D n ( 3 ) ( x 0 ) b m n i η r D n ( 1 ) ( x 2 ) D n ( 3 ) ( x 0 ) η r D n ( 1 ) ( x 1 ) D n ( 3 ) ( x 0 ) + D n ( 1 ) ( x 2 ) η r D n ( 3 ) ( x 0 ) D n ( 1 ) ( x 1 ) η r D n ( 3 ) ( x 0 ) .
A m n s = ψ n ( x 0 ) ξ n ( x 0 ) D n ( 1 ) ( x ) η r D n ( 1 ) ( x 0 ) D n ( 1 ) ( x ) η r D n ( 3 ) ( x 0 ) a m n i ,
B m n s = ψ n ( x 0 ) ξ n ( x 0 ) η r D n ( 1 ) ( x ) D n ( 1 ) ( x 0 ) η r D n ( 1 ) ( x ) D n ( 3 ) ( x 0 ) b m n i .
a m n i = { i n + 1 2 n + 1 2 n ( n + 1 ) m = 1 i n + 1 2 n + 1 2 m = 1 , b m n i = { i n + 1 2 n + 1 2 n ( n + 1 ) m = 1 i n + 1 2 n + 1 2 m = 1 ,
E θ s = E 0 exp ( i k r ) k r n = 1 m = n n ( i ) n [ A m n s m π m n + B m n s τ m n ] e i m ϕ ,
E ϕ s = i E 0 exp ( i k r ) k r n = 1 m = n n ( i ) n [ A m n s τ m n + B m n s m π m n ] e i m ϕ ,
π m n = P n m ( cos θ ) sin θ , τ m n = d P n m ( cos θ ) d θ .
E θ s = i E 0 exp ( i k r ) k r n = 1 2 n + 1 n ( n + 1 ) [ cos ϕ ( A n s a π n + B n s b τ n ) + i sin ϕ ( A n s b π n + B n s a τ n ) ] ,
E ϕ s = E 0 exp ( i k r ) k r n = 1 2 n + 1 n ( n + 1 ) [ cos ϕ ( B n s a π n + A n s b τ n ) + i sin ϕ ( B n s b π n + A n s a τ n ) ] ,
π n = π 1 n = P n 1 ( cos θ ) sin θ , τ n = τ 1 n = d P n 1 ( cos θ ) d θ .
σ θ = lim r 4 π r 2 | E θ s | 2 | E i | 2 , σ ϕ = lim r 4 π r 2 | E ϕ s | 2 | E i | 2 .
E θ s = E 0 exp ( i k r ) k r n = 1 m = n n ( i ) n [ ( A n s a a m n i + A n s b b m n i ) m π m n + ( B n s a a m n i + B n s b b m n i ) τ m n ] e i m ϕ = E 0 exp ( i k r ) k r n = 1 ( i ) n { e i ϕ [ ( A n s a a 1 n i + A n s b b 1 n i ) π 1 n + ( B n s a a 1 n i + B n s b b 1 n i ) τ 1 n ] + e i ϕ [ ( A n s a a 1 n i + A n s b b 1 n i ) π 1 n + ( B n s a a 1 n i + B n s b b 1 n i ) τ 1 n ] } ,
E ϕ s = i E 0 exp ( i k r ) k r n = 1 m = n n ( i ) n [ ( A n s a a m n i + A n s b b m n i ) τ m n + ( B n s a a m n i + B n s b b m n i ) m π m n ] e i m ϕ = i E 0 exp ( i k r ) k r n = 1 ( i ) n { e i ϕ [ ( A n s a a 1 n i + A n s b b 1 n i ) τ 1 n + ( B n s a a 1 n i + B n s b b 1 n i ) π 1 n ] + e i ϕ [ ( A n s a a 1 n i + A n s b b 1 n i ) τ 1 n ( B n s a a 1 n i + B n s b b 1 n i ) π 1 n ] } .
π 1 n = π 1 n n ( n + 1 ) , τ 1 n = τ 1 n n ( n + 1 ) .
a 1 n i π 1 n = a 1 n i π 1 n = i n + 1 2 n + 1 2 n ( n + 1 ) π 1 n ,
a 1 n i τ 1 n = a 1 n i τ 1 n = i n + 1 2 n + 1 2 n ( n + 1 ) τ 1 n ,
b 1 n i π 1 n = b 1 n i π 1 n = i n + 1 2 n + 1 2 n ( n + 1 ) π 1 n ,
b 1 n i τ 1 n = b 1 n i τ 1 n = i n + 1 2 n + 1 2 n ( n + 1 ) τ 1 n .

Metrics