Abstract

Optical properties of metal nanowires and nanowire composite materials are studied. An incident electromagnetic wave can effectively couple to the propagating surface plasmon polariton (SPP) modes in metal nanowires resulting in very large local fields. The excited SPP modes depend on the structure of nanowires and their orientation with respect to incident radiation. A nanowire percolation composite is shown to have a broadband spectrum of localized plasmon modes. We also show that a composite of nanowires arranged into parallel pairs can act as a left-handed material with the effective magnetic permeability and dielectric permittivity both negative in the visible and near-infrared spectral ranges.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
    [CrossRef]
  2. K. B. Shelimov and M. Moskovits, “Composite Nanostructures Based on Template-Grown Boron Nitride Nanotubules,” Chemistry of Materials,  12, 250 (2000);
    [CrossRef]
  3. J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters 75, 367 (1999);
    [CrossRef]
  4. J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966 (2000);
    [CrossRef] [PubMed]
  5. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics Uspekhi 10, 509 (1968).
    [CrossRef]
  6. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000);
    [CrossRef] [PubMed]
  7. G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys.Rev.B67, 035109 (2003)
    [CrossRef]
  8. V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials 11, 65 (2002)
    [CrossRef]
  9. L.V. Panina, A.N. Grigorenko, and D.P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys.Rev.B 66, 155411 (2002)
    [CrossRef]
  10. E.M. Purcell and C.R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophysical Journal 186, 705 (1973)
    [CrossRef]
  11. J.D. JacksonClassical Electrodynamics, (J.Wiley&Sons, Inc, 1999)
  12. B.T. Draine, “Discrete dipole approximation and its application to interstellar graphite grains,” Astrophys.J. 333, 848 (1988)
    [CrossRef]
  13. V.A. Markel, “Antisymmetrical optical states,” J.Opt.Soc.Am. B121783 (1995)
  14. V.A. Markel, “Scattering of light from two interacting spherical particles,” J.Mod.Opt 39853 (1992)
    [CrossRef]
  15. B.T. Draine “The discrete dipole approximation for light scattering by irregular targets” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, Acad.Press (2000)
  16. M. Moskovits, private communication
  17. N. Yamamoto, K. Araya, M. Nakano, and F.J.Garsia de Abajo, “Direct imaging of plasmons in nanostructures”, annual OSA meeting 2002, Orlando, Florida
  18. D. Stauffer and A. AharonyIntroduction to percolation theory, (Taylor and Fransis, 1994)
  19. S. Ducourtieux, et al, “Near-field optical studies of semicontinuous metal films,” Phys.Rev.B64 165403 (2001)
    [CrossRef]
  20. V. M. Shalaev (editor) Optical Properties of Nanostructured Random Media, Topics in Applied Physics, v. 82, (Springer Verlag, Berlin, 2002)
    [CrossRef]
  21. A.N. Lagarkov and A.K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys.Rev.B 53, 6318 (1996)
    [CrossRef]

2002 (2)

V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials 11, 65 (2002)
[CrossRef]

L.V. Panina, A.N. Grigorenko, and D.P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys.Rev.B 66, 155411 (2002)
[CrossRef]

2000 (4)

D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000);
[CrossRef] [PubMed]

S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
[CrossRef]

K. B. Shelimov and M. Moskovits, “Composite Nanostructures Based on Template-Grown Boron Nitride Nanotubules,” Chemistry of Materials,  12, 250 (2000);
[CrossRef]

J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966 (2000);
[CrossRef] [PubMed]

1999 (1)

J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters 75, 367 (1999);
[CrossRef]

1996 (1)

A.N. Lagarkov and A.K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys.Rev.B 53, 6318 (1996)
[CrossRef]

1995 (1)

V.A. Markel, “Antisymmetrical optical states,” J.Opt.Soc.Am. B121783 (1995)

1992 (1)

V.A. Markel, “Scattering of light from two interacting spherical particles,” J.Mod.Opt 39853 (1992)
[CrossRef]

1988 (1)

B.T. Draine, “Discrete dipole approximation and its application to interstellar graphite grains,” Astrophys.J. 333, 848 (1988)
[CrossRef]

1973 (1)

E.M. Purcell and C.R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophysical Journal 186, 705 (1973)
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics Uspekhi 10, 509 (1968).
[CrossRef]

Aharony, A.

D. Stauffer and A. AharonyIntroduction to percolation theory, (Taylor and Fransis, 1994)

Araya, K.

N. Yamamoto, K. Araya, M. Nakano, and F.J.Garsia de Abajo, “Direct imaging of plasmons in nanostructures”, annual OSA meeting 2002, Orlando, Florida

Brown, S. D. M.

S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
[CrossRef]

Corio, P.

S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
[CrossRef]

de Abajo, F.J.Garsia

N. Yamamoto, K. Araya, M. Nakano, and F.J.Garsia de Abajo, “Direct imaging of plasmons in nanostructures”, annual OSA meeting 2002, Orlando, Florida

Draine, B.T.

B.T. Draine, “Discrete dipole approximation and its application to interstellar graphite grains,” Astrophys.J. 333, 848 (1988)
[CrossRef]

B.T. Draine “The discrete dipole approximation for light scattering by irregular targets” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, Acad.Press (2000)

Dresselhaus, G.

S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
[CrossRef]

Dresselhaus, M. S.

S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
[CrossRef]

Ducourtieux, S.

S. Ducourtieux, et al, “Near-field optical studies of semicontinuous metal films,” Phys.Rev.B64 165403 (2001)
[CrossRef]

Grigorenko, A.N.

L.V. Panina, A.N. Grigorenko, and D.P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys.Rev.B 66, 155411 (2002)
[CrossRef]

Jackson, J.D.

J.D. JacksonClassical Electrodynamics, (J.Wiley&Sons, Inc, 1999)

Lagarkov, A.N.

A.N. Lagarkov and A.K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys.Rev.B 53, 6318 (1996)
[CrossRef]

Li, J.

J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters 75, 367 (1999);
[CrossRef]

Makhnovskiy, D.P.

L.V. Panina, A.N. Grigorenko, and D.P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys.Rev.B 66, 155411 (2002)
[CrossRef]

Markel, V.A.

V.A. Markel, “Antisymmetrical optical states,” J.Opt.Soc.Am. B121783 (1995)

V.A. Markel, “Scattering of light from two interacting spherical particles,” J.Mod.Opt 39853 (1992)
[CrossRef]

Marucci, A.

S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
[CrossRef]

Moskovits, M.

K. B. Shelimov and M. Moskovits, “Composite Nanostructures Based on Template-Grown Boron Nitride Nanotubules,” Chemistry of Materials,  12, 250 (2000);
[CrossRef]

J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters 75, 367 (1999);
[CrossRef]

M. Moskovits, private communication

Nakano, M.

N. Yamamoto, K. Araya, M. Nakano, and F.J.Garsia de Abajo, “Direct imaging of plasmons in nanostructures”, annual OSA meeting 2002, Orlando, Florida

Nemat-Nasser, S.C.

D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000);
[CrossRef] [PubMed]

Padilla, W.J.

D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000);
[CrossRef] [PubMed]

Panina, L.V.

L.V. Panina, A.N. Grigorenko, and D.P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys.Rev.B 66, 155411 (2002)
[CrossRef]

Papadopoulos, C.

J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters 75, 367 (1999);
[CrossRef]

Pendry, J.B.

J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966 (2000);
[CrossRef] [PubMed]

Pennypacker, C.R.

E.M. Purcell and C.R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophysical Journal 186, 705 (1973)
[CrossRef]

Pimenta, M. A.

S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
[CrossRef]

Podolskiy, V.A.

V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials 11, 65 (2002)
[CrossRef]

Purcell, E.M.

E.M. Purcell and C.R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophysical Journal 186, 705 (1973)
[CrossRef]

Sarychev, A.K.

V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials 11, 65 (2002)
[CrossRef]

A.N. Lagarkov and A.K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys.Rev.B 53, 6318 (1996)
[CrossRef]

Shalaev, V.M.

V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials 11, 65 (2002)
[CrossRef]

Shelimov, K. B.

K. B. Shelimov and M. Moskovits, “Composite Nanostructures Based on Template-Grown Boron Nitride Nanotubules,” Chemistry of Materials,  12, 250 (2000);
[CrossRef]

Shultz, S.

D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000);
[CrossRef] [PubMed]

Shvets, G.

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys.Rev.B67, 035109 (2003)
[CrossRef]

Smith, D.R.

D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000);
[CrossRef] [PubMed]

Stauffer, D.

D. Stauffer and A. AharonyIntroduction to percolation theory, (Taylor and Fransis, 1994)

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics Uspekhi 10, 509 (1968).
[CrossRef]

Vier, D.C.

D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000);
[CrossRef] [PubMed]

Xu, J.M.

J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters 75, 367 (1999);
[CrossRef]

Yamamoto, N.

N. Yamamoto, K. Araya, M. Nakano, and F.J.Garsia de Abajo, “Direct imaging of plasmons in nanostructures”, annual OSA meeting 2002, Orlando, Florida

Applied Physics Letters (1)

J. Li, C. Papadopoulos, J.M. Xu, and M. Moskovits, “Highly-ordered carbon nanotube arrays for electronics applications,” Applied Physics Letters 75, 367 (1999);
[CrossRef]

Astrophys.J. (1)

B.T. Draine, “Discrete dipole approximation and its application to interstellar graphite grains,” Astrophys.J. 333, 848 (1988)
[CrossRef]

Astrophysical Journal (1)

E.M. Purcell and C.R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophysical Journal 186, 705 (1973)
[CrossRef]

Chemistry of Materials (1)

K. B. Shelimov and M. Moskovits, “Composite Nanostructures Based on Template-Grown Boron Nitride Nanotubules,” Chemistry of Materials,  12, 250 (2000);
[CrossRef]

J.Mod.Opt (1)

V.A. Markel, “Scattering of light from two interacting spherical particles,” J.Mod.Opt 39853 (1992)
[CrossRef]

J.Opt.Soc.Am. (1)

V.A. Markel, “Antisymmetrical optical states,” J.Opt.Soc.Am. B121783 (1995)

Journal of Nonlinear Optical Physics and Materials (1)

V.A. Podolskiy, A.K. Sarychev, and V.M. Shalaev, “Plasmon modes in metal nanowires and left-handed materials,” Journal of Nonlinear Optical Physics and Materials 11, 65 (2002)
[CrossRef]

Phys. Rev. B (1)

S. D. M. Brown, P. Corio, A. Marucci, M. A. Pimenta, M. S. Dresselhaus, and G. Dresselhaus, “Second-order resonant Raman spectra of single-walled carbon nanotubes,” Phys. Rev. B 61, 7734–7742 (2000);
[CrossRef]

Phys. Rev. Lett. (2)

J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966 (2000);
[CrossRef] [PubMed]

D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Shultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84, 4184 (2000);
[CrossRef] [PubMed]

Phys.Rev.B (2)

L.V. Panina, A.N. Grigorenko, and D.P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys.Rev.B 66, 155411 (2002)
[CrossRef]

A.N. Lagarkov and A.K. Sarychev, “Electromagnetic properties of composites containing elongated conducting inclusions,” Phys.Rev.B 53, 6318 (1996)
[CrossRef]

Soviet Physics Uspekhi (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics Uspekhi 10, 509 (1968).
[CrossRef]

Other (8)

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys.Rev.B67, 035109 (2003)
[CrossRef]

J.D. JacksonClassical Electrodynamics, (J.Wiley&Sons, Inc, 1999)

B.T. Draine “The discrete dipole approximation for light scattering by irregular targets” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, Acad.Press (2000)

M. Moskovits, private communication

N. Yamamoto, K. Araya, M. Nakano, and F.J.Garsia de Abajo, “Direct imaging of plasmons in nanostructures”, annual OSA meeting 2002, Orlando, Florida

D. Stauffer and A. AharonyIntroduction to percolation theory, (Taylor and Fransis, 1994)

S. Ducourtieux, et al, “Near-field optical studies of semicontinuous metal films,” Phys.Rev.B64 165403 (2001)
[CrossRef]

V. M. Shalaev (editor) Optical Properties of Nanostructured Random Media, Topics in Applied Physics, v. 82, (Springer Verlag, Berlin, 2002)
[CrossRef]

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 (9)

Fig. 1.
Fig. 1.

A nanowire represented by an array of “intersecting” spheres [8]

Fig. 2.
Fig. 2.

Intensity enhancement distribution around silver nanowire excited by a plane electromagnetic wave [8]. The wavelength of incident light is 540 nm. The angle between the needle and the light wavevector is 30°, the wavevector and vector E of the incident irradiation are in the plane of the figure

Fig. 3.
Fig. 3.

Surface plasmon polariton resonance in a silver nanowire excited by a plane electromagnetic wave [8]. The intensity distribution (top panel) and simulated near-field optical microscope image (bottom panel) are shown. The wavelength of incident light is 540 nm; the angle between the nanowire and the wavevector of the incident light is 30°. The wavevector and E vector of the incident irradiation are in the plane of the figure; the needle length is 480 nm

Fig. 4.
Fig. 4.

Nanowire percolation Ag composite (left) and the field distribution over this composite for the incident wavelength of 550 nm (center) and 750 nm (right). In both figures the case of normal incidence with Ex is considered [8]

Fig. 5.
Fig. 5.

Two parallel nanowires (a) and a layer of such pairs (b). A composite material based on such nanowire pairs may have the negative refraction index in the optical range [8]

Fig. 6.
Fig. 6.

Numerically simulated dielectric (left panels) and magnetic (right panels) moments for single nanowires (b 1=0.35µm, b 2=0.05µm) (green lines) and their pairs (d=0.15µm) (red lines) compared to the analytical Eqs. (11) (black lines). Magnetic moment of the single nanowire is multiplied by 10. The moments are normalized to the unit volume.

Fig. 7.
Fig. 7.

Dielectric moments for individual nanowires (left) and for nanowire pairs (center) and magnetic moments for nanowire pairs as functions of wavelengths. The nanowire thickness is varied for different plots: b 2=0.035µm (blue), b 2=0.05µm (red), and b 2=0.07µm (green); for all plots, b 1=0.35µm and d=0.23µm. The moments are normalized to the unit volume

Fig. 8.
Fig. 8.

Dielectric (left) and magnetic (right) moments in nanowire pairs as functions of wavelengths. The distance between the nanowires in the pairs is varied: d=0.15µm (red), d=0.23µm(blue), d=0.3µm (green), and d=0.45µm (black)); for all plots, b 1=0.35µm and b 2=0.05µm. The moments are normalized to the unit volume

Fig. 9.
Fig. 9.

Real (colored) and imaginary (black) parts of dielectric (left) and magnetic (right) moments in nanowire pair as functions of wavelengths. The system parameters: d=0.075µm, b 1=0.35µm and b 2=0.025µm. The moments are normalized to the unit volume

Equations (14)

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

d i = α 0 [ E i n c + j i N G ̂ ( r i r j ) d j ] ,
G α β = k 3 [ A ( k r ) δ α β + B ( k r ) r α r β ] ,
A ( x ) = [ x 1 + i x 2 x 3 ] exp ( i x ) ,
B ( x ) = [ x 1 3 i x 2 + 3 x 3 ] exp ( i x ) ,
α 0 = α L L 1 i ( 2 k 3 3 ) α L L
α L L = R 3 1 + 2 ,
a R = ( 4 π 3 ) 1 3 1.612
A = e i k R c R [ e i k 2 ( n · d ) b 1 b 1 e i k 2 ( n · ρ ) j 1 ( ρ ) d ρ + e i k 2 ( n · d ) b 1 b 1 e i k n · ρ j 2 ( ρ ) d ρ ] ,
A = e i k R c R [ b 1 b 1 ( j 1 + j 2 ) d ρ i k 2 ( n · d ) b 1 b 1 ( j 1 j 2 ) d ρ ] .
P = p ( r ) d r ,
A m q = i k e i k R R [ [ n × M ] + d 2 c b 1 b 2 ( n · ( j 1 j 2 ) ) d ρ ] ,
M = 1 2 c [ j ( r ) × r ] d r ,
M = 2 H b 1 3 C 2 ( k d ) 2 tan ( g b 1 ) g b 1 ( g b 1 ) 3
P = 2 3 b 1 b 2 2 f ( Δ ) E m 1 1 + f ( Δ ) m ( b 1 b 2 ) 2 ln ( 1 + b 1 b 2 ) cos Ω ,

Metrics