Abstract

The Lorenz-Mie theory is revisited to explicitly include materials whose permeability is different from unity. The expansion coefficients of the scattered field are given for light scattering by both homogeneous and coated spheres. It is shown that the backscatter is exactly zero if the impedance of the spherical particles is equal to the intrinsic impedance of the surrounding medium. If spherical particles are sufficiently large, the zero backscatter can be explained as impedance matching using the asymptotic expression for the radar backscattering cross section. In the case of a coated sphere, the shell can be regarded as a cloak if the product of the thickness and the imaginary part of the refractive index of the outer shell is large.

© 2008 Optical Society of America

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  3. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
    [Crossref] [PubMed]
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    [Crossref]
  5. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
    [Crossref] [PubMed]
  6. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621-1–5 (2006).
  7. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
    [Crossref] [PubMed]
  8. H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903-1–4 (2007).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2007 (3)

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[Crossref] [PubMed]

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903-1–4 (2007).
[Crossref] [PubMed]

B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101-1–4 (R) (2007)
[Crossref]

2006 (6)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. London, Ser. A 462, 3027–3059 (2006).
[Crossref]

J.R. Liu, M. Itoh, and K.-I Machida, “Frequency dispersion of complex permeability and permittivity on ironbased nanocomposites derived from rare earth-iron intermetallic compounds,” J. Alloys Compd. 408–412, 1396–1399 (2006).
[Crossref]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

2005 (1)

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623-1–9 (2005).
[Crossref]

2004 (1)

1996 (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in maxwells equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

1981 (1)

1980 (1)

1908 (1)

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig)  25, 377–455 (1908).
[Crossref]

1904 (1)

J. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385 (1904).
[Crossref]

Ackerman, T. P.

Alu, A.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623-1–9 (2005).
[Crossref]

Bohren, C. F.

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

Chen, H.

B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101-1–4 (R) (2007)
[Crossref]

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903-1–4 (2007).
[Crossref] [PubMed]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621-1–5 (2006).

Engheta, N.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623-1–9 (2005).
[Crossref]

Garnett, J. Maxwell

J. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385 (1904).
[Crossref]

Guenneau, S.

Huffman, D. R.

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

Itoh, M.

J.R. Liu, M. Itoh, and K.-I Machida, “Frequency dispersion of complex permeability and permittivity on ironbased nanocomposites derived from rare earth-iron intermetallic compounds,” J. Alloys Compd. 408–412, 1396–1399 (2006).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd Edition (Wiley-VCH, 1998),

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Kattawar, G. W.

Kong, J. A.

B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101-1–4 (R) (2007)
[Crossref]

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903-1–4 (2007).
[Crossref] [PubMed]

Lee, Y. -K.

Leonhardt, U.

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

Liu, J.R.

J.R. Liu, M. Itoh, and K.-I Machida, “Frequency dispersion of complex permeability and permittivity on ironbased nanocomposites derived from rare earth-iron intermetallic compounds,” J. Alloys Compd. 408–412, 1396–1399 (2006).
[Crossref]

Luo, Y.

B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101-1–4 (R) (2007)
[Crossref]

Machida, K.-I

J.R. Liu, M. Itoh, and K.-I Machida, “Frequency dispersion of complex permeability and permittivity on ironbased nanocomposites derived from rare earth-iron intermetallic compounds,” J. Alloys Compd. 408–412, 1396–1399 (2006).
[Crossref]

Mie, G.

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig)  25, 377–455 (1908).
[Crossref]

Milton, G. W.

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. London, Ser. A 462, 3027–3059 (2006).
[Crossref]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Nicolet, A.

Nicorovici, N.-A. P.

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. London, Ser. A 462, 3027–3059 (2006).
[Crossref]

Pendry, J. B.

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
[Crossref] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

A. J. Ward and J. B. Pendry, “Refraction and geometry in maxwells equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621-1–5 (2006).

Popa, B.-I.

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621-1–5 (2006).

Ran, L.

B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101-1–4 (R) (2007)
[Crossref]

Schurig, D.

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621-1–5 (2006).

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621-1–5 (2006).

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Toon, O. B.

van de Hulst, H.C.

H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in maxwells equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

Wiscombe, W. J.

W. J. Wiscombe, “Mie scattering calculation,” NCAR Tech. Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).

Wiscombe, W.J.

Wu, B.

B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101-1–4 (R) (2007)
[Crossref]

Wu, B. I.

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903-1–4 (2007).
[Crossref] [PubMed]

Yang, P.

Zhai, P. -W.

Zhang, B.

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903-1–4 (2007).
[Crossref] [PubMed]

B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101-1–4 (R) (2007)
[Crossref]

Zolla, F.

Ann. Phys. (1)

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig)  25, 377–455 (1908).
[Crossref]

Appl. Opt. (3)

J. Alloys Compd. (1)

J.R. Liu, M. Itoh, and K.-I Machida, “Frequency dispersion of complex permeability and permittivity on ironbased nanocomposites derived from rare earth-iron intermetallic compounds,” J. Alloys Compd. 408–412, 1396–1399 (2006).
[Crossref]

J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in maxwells equations,” J. Mod. Opt. 43, 773–793 (1996).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Philos. Trans. R. Soc. London (1)

J. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385 (1904).
[Crossref]

Phys. Rev. B (1)

B. Zhang, H. Chen, B. Wu, Y. Luo, L. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Phys. Rev. B 76, 121101-1–4 (R) (2007)
[Crossref]

Phys. Rev. E (2)

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623-1–9 (2005).
[Crossref]

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621-1–5 (2006).

Phys. Rev. Lett. (1)

H. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic Wave Interactions with a Metamaterial Cloak,” Phys. Rev. Lett. 99, 063903-1–4 (2007).
[Crossref] [PubMed]

Proc. R. Soc. London, Ser. A (1)

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. London, Ser. A 462, 3027–3059 (2006).
[Crossref]

Science (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Other (4)

H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

J. D. Jackson, Classical Electrodynamics, 3rd Edition (Wiley-VCH, 1998),

W. J. Wiscombe, “Mie scattering calculation,” NCAR Tech. Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).

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Figures (3)

Fig. 1.
Fig. 1.

The phase function as a function of the scattering angle for homogeneous spherical particles. η=1 and the size parameter x=10 are used. The refractive indices plotted are m=1.1,1.3,1.7,1.1+0.1i,1.3+0.1i,1.7+0.1i

Fig. 2.
Fig. 2.

The phase function as a function of the scattering angle for homogeneous spherical particles. The size parameter is x=10. The refractive indices is m=1.3. η=0.9999,0.999,0.99,0.91 are plotted.

Fig. 3.
Fig. 3.

The phase function as a function of the scattering angle for coated spherical particles. The size of the inner sphere is x=5. and m1=1.5, η 1=2/3. Two refractive indices for the outer sphere are chosen: m2=2+0.5i and m2=2+i. η 2=1. For each of the refractive indices, a proper value of y is used to produce τ=1 and τ=4 according to τ=(y-x)Im(m2).

Tables (3)

Tables Icon

Table 1. Extinction and scattering efficiencies for the cases in Fig. 1

Tables Icon

Table 2. Extinction/scattering efficiencies for cases in Fig. 2

Tables Icon

Table 3. Extinction and scattering efficiencies for cases in Fig. 3. Also shown are the ratios of the extinction and scattering cross sections of the coated particles and those of the “Not coated” case.

Equations (20)

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ε = = ε r ( r ) r ̂ r ̂ + ε t ( θ ̂ θ ̂ + ϕ ̂ ϕ ̂ ) ,
μ = = μ r ( r ) r ̂ r ̂ + μ t ( θ ̂ θ ̂ + ϕ ̂ ϕ ̂ ) ,
η i = μ i ε i , i = 1 , 0 ,
x = ka , m = ε 1 μ 1 ε 0 μ 0 ,
η = η 1 η 0 = μ 1 ε 1 μ 0 ε 0 .
a n = [ D n ( m x ) η + n x ] ψ n ( x ) ψ n 1 ( x ) [ D n ( m x ) η + n x ] ξ n ( x ) ξ n 1 ( x ) ,
b n = [ D n ( m x ) η + n x ] ψ n ( x ) ψ n 1 ( x ) [ D n ( m x ) η + n x ] ξ n ( x ) ξ n 1 ( x ) ,
S 1 = S 2 = n 2 n + 1 n ( n + 1 ) a n ( π n + τ n ) ,
lim x Q b = R ( 0 o ) ,
S 11 = S 2 2 + S 1 2 2 , S 12 = S 2 2 S 1 2 2 , S 33 = S 2 * S 1 + S 2 S 1 * 2 , S 34 = i ( S 1 S 2 * S 2 S 1 * ) 2 .
x = ka , m 1 = ε 1 μ 1 ε 3 μ 3 , η 1 = μ 1 ε 1 μ 3 ε 3 ,
y = kb , m 2 = ε 2 μ 2 ε 3 μ 3 , η 2 = μ 2 ε 2 μ 3 ε 3 ,
a n = [ η 2 D ˜ n + n y ] ψ n ( y ) ψ n 1 ( y ) [ η 2 D ˜ n + n y ] ξ n ( y ) ξ n 1 ( y ) ,
b n = [ G ˜ n η 2 + n y ] ψ n ( y ) ψ n 1 ( y ) [ G ˜ n η 2 + n y ] ξ n ( y ) ξ n 1 ( y ) ,
D ˜ n = D n ( m 2 y ) A n χ n ( m 2 y ) ψ n ( m 2 y ) 1 A n χ n ( m 2 y ) ψ n ( m 2 y ) ,
G ˜ n = D n ( m 2 y ) B n χ n ( m 2 y ) ψ n ( m 2 y ) 1 B n χ n ( m 2 y ) ψ n ( m 2 y ) ,
A n = ψ n ( m 2 x ) η 1 D n ( m 1 x ) η 2 D n ( m 2 x ) η 1 D n ( m 1 x ) χ n ( m 2 x ) η 2 χ n ( m 2 x ) ,
B n = ψ n ( m 2 x ) η 1 D n ( m 2 x ) η 2 D n ( m 1 x ) η 1 χ n ( m 2 x ) η 2 D n ( m 1 x ) χ n ( m 2 x ) .
η 2 = 1 , τ = k ( b a ) Im ( m 2 ) = ( y x ) Im ( m 2 ) 1 ,
C ext , sca C ext , sca Nc = Q ext , sca · y 2 C ext , sca Nc · x 2 ,

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