Abstract

Lorenz–Mie resonances produced by small spheres are analyzed as a function of their size and optical properties (ε0, μ0). New generalized (μ1) approximate and compact expressions of the first four Lorenz–Mie coefficients (a1, b1, a2, and b2) are calculated. With these expressions and for small particles with various values of ε and μ, the extinction cross section (Qext) is calculated and analyzed, in particular for resonant conditions. The dependence on particle size of the extinction resonance, together with the resonance shape (FWHM), is also analyzed. In addition to the former analysis, a study of the scattering diagrams for some interesting values of ε and μ is also presented.

© 2008 Optical Society of America

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References

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    [CrossRef]
  2. L. Lorenz, Oeuvres Scientifiques (Johnson, 1964).
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  7. H. M. Hiep, T. Endo, K. Kerman, M. Chikae, D.-K. Kim, S. Yamamura, Y. Takamura, and E. Tamiya, "A localized surface plasmon resonance based immunosensor for the detection of casein in milk," Sci. Technol. Adv. Mater. 8, 331-338 (2007).
    [CrossRef]
  8. C. Guo, P. Boullanger, L. Jiang, and T. Liu, "Highly sensitive gold nanoparticle biosensor chips modified with a self-assembled bilayer for detection of Con A," Biosens. Bioelectron. 22, 1830-1834 (2007).
    [CrossRef]
  9. S. Pillai, K. R. Cathpole, T. Trupke, and M. A. Green, "Surface plasmon enhanced silicon solar cells," J. Appl. Phys. 101, 093105 (2007).
    [CrossRef]
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    [CrossRef]
  12. S. C. Hill and R. E. Benner, "Morphology-dependent resonances," in Optical Effects Associated with Small Particles, P.W.Barber and R.K.Chang, eds. (World Scientific, 1988).
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    [CrossRef]
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    [CrossRef]
  15. G. Roll and G. Schweiger, "Geometrical optics model of Mie resonances," J. Opt. Soc. Am. A 17, 1301-1311 (2000).
    [CrossRef]
  16. B. M. Reinhard, M. Siu, H. Agarwal, A. P. Alivisatos, and J. Liphardt, "Calibration of a dynamic molecular ruler based on plasmon coupling between gold nanoparticles," Nano Lett. 5, 2246-2252 (2005).
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    [CrossRef] [PubMed]
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    [CrossRef]
  24. R. Ruppin, "Extinction properties of a sphere with negative permittivity and permeability," Solid State Commun. 116, 411-415 (2000).
    [CrossRef]
  25. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nassser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
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    [CrossRef]
  27. R. Ruppin, "Surface polaritons and extinction properties of a left-handed material cylinder," J. Phys.: Condens. Matter 16, 5991-5998 (2004).
    [CrossRef]
  28. A. Alú, A. Salandrino, and N. Engheta, "Negative effective permeability and left-handed materials at optical frequencies," Opt. Express 14, 1557-1567 (2006).
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    [CrossRef]
  31. N. Engheta, "Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials," Science 317, 1698-1702 (2007).
    [CrossRef] [PubMed]

2007

H. M. Hiep, T. Endo, K. Kerman, M. Chikae, D.-K. Kim, S. Yamamura, Y. Takamura, and E. Tamiya, "A localized surface plasmon resonance based immunosensor for the detection of casein in milk," Sci. Technol. Adv. Mater. 8, 331-338 (2007).
[CrossRef]

C. Guo, P. Boullanger, L. Jiang, and T. Liu, "Highly sensitive gold nanoparticle biosensor chips modified with a self-assembled bilayer for detection of Con A," Biosens. Bioelectron. 22, 1830-1834 (2007).
[CrossRef]

S. Pillai, K. R. Cathpole, T. Trupke, and M. A. Green, "Surface plasmon enhanced silicon solar cells," J. Appl. Phys. 101, 093105 (2007).
[CrossRef]

O. Merchiers, F. Moreno, F. González, and J. M. Saiz, "Light scattering by an ensemble of interacting dipolar particles with both electric and magnetic polarizabilities," Phys. Rev. A 76, 043834 (2007).
[CrossRef]

O. Merchiers, F. Moreno, F. González, J. M. Saiz, and G. Videen, "Electromagnetic wave scattering from two interacting small spherical particles. Influence of their optical constants, epsi and μ," Opt. Commun. 269, 1-7 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelength," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

V. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

N. Engheta, "Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials," Science 317, 1698-1702 (2007).
[CrossRef] [PubMed]

2006

2005

B. M. Reinhard, M. Siu, H. Agarwal, A. P. Alivisatos, and J. Liphardt, "Calibration of a dynamic molecular ruler based on plasmon coupling between gold nanoparticles," Nano Lett. 5, 2246-2252 (2005).
[CrossRef] [PubMed]

2004

R. Ruppin, "Surface polaritons and extinction properties of a left-handed material cylinder," J. Phys.: Condens. Matter 16, 5991-5998 (2004).
[CrossRef]

2002

V. Kuzmiak and A. A. Maradudin, "Scattering properties of a cylinder fabricated from a left-handed material," Phys. Rev. B 66, 045116 (2002).
[CrossRef]

2000

R. Ruppin, "Extinction properties of a sphere with negative permittivity and permeability," Solid State Commun. 116, 411-415 (2000).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nassser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

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

G. Roll and G. Schweiger, "Geometrical optics model of Mie resonances," J. Opt. Soc. Am. A 17, 1301-1311 (2000).
[CrossRef]

1999

T. Jensen, L. Kelly, A. Lazarides, and G. Schatz, "Electrodynamics of noble metal nanoparticles and nanoparticles clusters," J. Cluster Sci. 10, 295-317 (1999).
[CrossRef]

1994

G. W. Mulholland, C. F. Bohren, and K. A. Fuller, "Light scattering by agglomerates: coupled electric and magnetic dipole method," Langmuir 10, 2533-2546 (1994).
[CrossRef]

1992

G. Videen and W. S. Bickel, "Light scattering resonances in small spheres," Phys. Rev. A 45, 6008-6012 (1992).
[CrossRef] [PubMed]

1983

1976

1975

R. Ruppin, "Optical properties of small metal spheres," Phys. Rev. B 11, 2871-2876 (1975).
[CrossRef]

1908

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Ann. Phys.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Biosens. Bioelectron.

M. Kreuzer, R. Quidant, G. Badenes, and M. P. Marco, "Quantitative detection of doping substances by a localised surface plasmon sensor," Biosens. Bioelectron. 21, 1345-1349 (2006).
[CrossRef]

C. Guo, P. Boullanger, L. Jiang, and T. Liu, "Highly sensitive gold nanoparticle biosensor chips modified with a self-assembled bilayer for detection of Con A," Biosens. Bioelectron. 22, 1830-1834 (2007).
[CrossRef]

J. Appl. Phys.

S. Pillai, K. R. Cathpole, T. Trupke, and M. A. Green, "Surface plasmon enhanced silicon solar cells," J. Appl. Phys. 101, 093105 (2007).
[CrossRef]

J. Cluster Sci.

T. Jensen, L. Kelly, A. Lazarides, and G. Schatz, "Electrodynamics of noble metal nanoparticles and nanoparticles clusters," J. Cluster Sci. 10, 295-317 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys.: Condens. Matter

R. Ruppin, "Surface polaritons and extinction properties of a left-handed material cylinder," J. Phys.: Condens. Matter 16, 5991-5998 (2004).
[CrossRef]

Langmuir

G. W. Mulholland, C. F. Bohren, and K. A. Fuller, "Light scattering by agglomerates: coupled electric and magnetic dipole method," Langmuir 10, 2533-2546 (1994).
[CrossRef]

Nano Lett.

B. M. Reinhard, M. Siu, H. Agarwal, A. P. Alivisatos, and J. Liphardt, "Calibration of a dynamic molecular ruler based on plasmon coupling between gold nanoparticles," Nano Lett. 5, 2246-2252 (2005).
[CrossRef] [PubMed]

Nat. Photonics

V. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Commun.

O. Merchiers, F. Moreno, F. González, J. M. Saiz, and G. Videen, "Electromagnetic wave scattering from two interacting small spherical particles. Influence of their optical constants, epsi and μ," Opt. Commun. 269, 1-7 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

O. Merchiers, F. Moreno, F. González, and J. M. Saiz, "Light scattering by an ensemble of interacting dipolar particles with both electric and magnetic polarizabilities," Phys. Rev. A 76, 043834 (2007).
[CrossRef]

G. Videen and W. S. Bickel, "Light scattering resonances in small spheres," Phys. Rev. A 45, 6008-6012 (1992).
[CrossRef] [PubMed]

Phys. Rev. B

R. Ruppin, "Optical properties of small metal spheres," Phys. Rev. B 11, 2871-2876 (1975).
[CrossRef]

V. Kuzmiak and A. A. Maradudin, "Scattering properties of a cylinder fabricated from a left-handed material," Phys. Rev. B 66, 045116 (2002).
[CrossRef]

Phys. Rev. Lett.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nassser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

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

Sci. Technol. Adv. Mater.

H. M. Hiep, T. Endo, K. Kerman, M. Chikae, D.-K. Kim, S. Yamamura, Y. Takamura, and E. Tamiya, "A localized surface plasmon resonance based immunosensor for the detection of casein in milk," Sci. Technol. Adv. Mater. 8, 331-338 (2007).
[CrossRef]

Science

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelength," Science 315, 47-49 (2007).
[CrossRef] [PubMed]

N. Engheta, "Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials," Science 317, 1698-1702 (2007).
[CrossRef] [PubMed]

Solid State Commun.

R. Ruppin, "Extinction properties of a sphere with negative permittivity and permeability," Solid State Commun. 116, 411-415 (2000).
[CrossRef]

Other

L. Lorenz, Oeuvres Scientifiques (Johnson, 1964).

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

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

P. N. Prasad, Nanophotonics (Wiley-Interscience, 2004).
[CrossRef]

S. C. Hill and R. E. Benner, "Morphology-dependent resonances," in Optical Effects Associated with Small Particles, P.W.Barber and R.K.Chang, eds. (World Scientific, 1988).

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Figures (8)

Fig. 1
Fig. 1

Comparative plot of Q ext for three different expressions of Mie coefficients: Exact (solid curve), approximate with Eqs. (5, 6) (AC1), and approximate using more coefficients in the expansion of sine and cosine (AC2) for μ = 1 . (a) Metallic case ( ε < 0 ) , (b) dielectric case ( ε > 0 ) .

Fig. 2
Fig. 2

3D plots of log ( Q ext ) as a function of the optical properties (ε and μ) for a spherical particle of R = 0.01 λ .

Fig. 3
Fig. 3

Enlargement of two interesting zones of Fig. 2: (a) Region with ε > 0 , μ > 1 ; (b) region with ε < 0 , μ < 0 . The coefficient that takes the highest value when this resonance is excited is indicated.

Fig. 4
Fig. 4

3D plots of log ( Q ext ) as a function of the optical properties (ε and μ) for two different ranges: (a) ε < 0 and μ > 0 , (b) ε > 0 and μ < 0 , when the range of values of ε and μ is equal.

Fig. 5
Fig. 5

Evolution of the extinction cross section as a function of the electric permittivity in the range ε < 0 for different values of particle radius R. The value of the magnetic permeability is indicated in the bottom right corner. Resonances are labeled with the Lorenz–Mie coefficient that takes the highest value at this point.

Fig. 6
Fig. 6

Evolution of FWHM and position of the resonances as a function of particle size.

Fig. 7
Fig. 7

Normalized scattering diagrams when electric dipolar (a),(b) and electric quadrupolar (c),(d) resonance is excited for different particle size. (a), (c) Figures correspond to TE incident polarization; (b) (d) to TM polarization.

Fig. 8
Fig. 8

Normalized scattering diagrams when magnetic dipolar (a),(b) and magnetic quadrupolar (c),(d) resonance is excited for different particle size. (a), (c) Figures correspond to TE incident polarization; (b), (d) to TM polarization. The scale shown in (b) is for a better visualization of the curves.

Tables (2)

Tables Icon

Table 1 Maximum Intensity of Light Scattered by a Small Sphere as a Function of Size of the Sphere and for Both Polarizations of the Incident Light When an Electric Resonance Is Excited a

Tables Icon

Table 2 Maximum Intensity of Light Scattered by a Small Sphere as a Function of Size of the Sphere and for both Polarizations of the Incident Light When a Magnetic Resonance Is Excited a

Equations (10)

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E s ( r ) = n = 1 E n [ i a n N e l n ( 3 ) ( k r ) b n M o l n ( 3 ) ( k r ) ] ,
H s ( r ) = ε μ n = 1 E n [ i b n N o l n ( 3 ) ( k r ) a n M e l n ( 3 ) ( k r ) ] .
a n = m ̃ ψ n ( x ) ψ n ( m x ) ψ n ( m x ) ψ n ( x ) m ̃ ξ n ( x ) ψ n ( m x ) ψ n ( m x ) ξ n ( x ) ,
b n = m ̃ ψ n ( x ) ψ n ( m x ) ψ n ( m x ) ψ n ( x ) m ̃ ψ n ( x ) ξ n ( m x ) ξ n ( m x ) ψ n ( x ) ,
Q ext = 2 x 2 n ( 2 n + 1 ) Re ( a n + b n ) .
a 1 2 i 3 x 3 ε 1 ε + 2 , b 1 2 i 3 x 3 μ 1 μ + 2 , a n b n 0 for n > 1 .
a 1 = m ̃ m x 3 [ m x cos ( m x ) sin ( m x ) ] cos ( m x ) ( m ̃ m 2 x 4 i m ̃ m 2 x + i m x 3 + i m x ) + sin ( m x ) ( m ̃ m x 3 + i m ̃ m i x 2 + i m 2 x 4 i + i m 2 x 2 ) ,
a 2 = cos ( m x ) ( 6 m ̃ m 2 x 2 6 m x 2 + m 3 x 4 ) + sin ( m x ) ( 6 x 3 m 2 x 3 6 m ̃ m x + 2 m ̃ m 3 x 3 ) { cos ( m x ) [ 3 i m ̃ m 3 x 2 + 2 m ̃ m 3 x 3 i m ̃ m 3 x 4 + 6 i m ̃ m 3 9 i m ̃ m 6 m ̃ m x + 3 i m ̃ m x 2 + 18 i m ̃ m x 2 ( 3 i x 2 2 i x ) ( 6 3 m 2 x 2 ) ] + sin ( m x ) [ 9 i m ̃ m 2 x + 6 m ̃ m 2 x 2 3 i m ̃ m 2 x 3 + 18 i m ̃ m 2 x ( 3 i x 2 2 i x ) ( 6 m x + m 3 x 3 ) ] } .
b n ( 1 m ̃ , m , x ) = a n ( m ̃ , m , x ) .
Q ext = 2 x 2 [ 3 Re ( a 1 + b 1 ) + 5 Re ( a 2 + b 2 ) ] .

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