Abstract

The Discrete Dipole Approximation (DDA) formalism has been generalized to materials with permeabilities µ ≠ 1 to study the scattering properties of impedance-matched aspherical particles and cloaked spheres. We have shown analytically that any impedance-matched particle with a four-fold rotational symmetry with respect to the direction of the incident radiation has the feature of zero backscatter. Moreover, an impedance-matched coat with the aforementioned symmetry property acting on an irregular dielectric particle with the same symmetry property can substantially reduce the backscatter. This leads to a substantial reduction of the signals from an object being detected by a monostatic radar/lidar system. The DDA simulation also provides accurate information about electric field distributions in the vicinity of a cloaked sphere.

© 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]
  2. 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]
  3. 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 (2006).
    [CrossRef]
  4. 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]
  5. H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99,063903 (2007).
    [CrossRef] [PubMed]
  6. A. Al`u and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72,016623 (2005).
    [CrossRef]
  7. P.-W. Zhai, Y. You, G. W. Kattawar, and P. Yang, "Monostatic Lidar/Radar invisibility using coated spheres," Opt. Express 16,1431-1439 (2008).
    [CrossRef] [PubMed]
  8. E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186,705-714 (1973).
    [CrossRef]
  9. B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333,848-872 (1988).
    [CrossRef]
  10. B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A 11,1491-1499 (1994).
    [CrossRef]
  11. J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, New York, 1975).
  12. A. Lakhtakia, "General theory of the Purcell-Pennypacker scattering approach and its extension to bianisotropic scatterers," Astrophys. J. 394,494-499 (1992).
    [CrossRef]
  13. B. T. Draine and J. Goodman, "Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405,685-697 (1993).
    [CrossRef]

2008 (1)

2007 (1)

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99,063903 (2007).
[CrossRef] [PubMed]

2006 (4)

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 (2006).
[CrossRef]

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]

2005 (1)

A. Al`u and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72,016623 (2005).
[CrossRef]

1994 (1)

1993 (1)

B. T. Draine and J. Goodman, "Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405,685-697 (1993).
[CrossRef]

1992 (1)

A. Lakhtakia, "General theory of the Purcell-Pennypacker scattering approach and its extension to bianisotropic scatterers," Astrophys. J. 394,494-499 (1992).
[CrossRef]

1988 (1)

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333,848-872 (1988).
[CrossRef]

1973 (1)

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186,705-714 (1973).
[CrossRef]

Astrophys. J. (4)

A. Lakhtakia, "General theory of the Purcell-Pennypacker scattering approach and its extension to bianisotropic scatterers," Astrophys. J. 394,494-499 (1992).
[CrossRef]

B. T. Draine and J. Goodman, "Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405,685-697 (1993).
[CrossRef]

E. M. Purcell and C. R. Pennypacker, "Scattering and absorption of light by nonspherical dielectric grains," Astrophys. J. 186,705-714 (1973).
[CrossRef]

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333,848-872 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (2)

Phys. Rev. E (2)

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 (2006).
[CrossRef]

A. Al`u and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72,016623 (2005).
[CrossRef]

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 (2007).
[CrossRef] [PubMed]

Science (2)

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]

Other (1)

J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, New York, 1975).

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Figures (7)

Fig. 1.
Fig. 1.

Operations on a four-fold symmetric particle. 1, 2, 3, and 4 are a group of points that rotate into each other. E and H are the incident electric and magnetic fields. The operation from (I) to (II) is a π/2 rotation of the coordinate system; the operation from (II) to (III) is a relabeling of E and H; (III) and (I) are identical except for a rotation of the four points. The coordinate system is as shown at the lower left corner, with the z-axis pointing out from the paper.

Fig. 2.
Fig. 2.

Comparison of P11 for coated spheres computed from DDA and Mie methods. Two cases, x = 2.5, y = 5, and x = 4, y = 8 were simulated. For the inner core, n1 = 1.5 and Z1 = 1/1.5, for the outer coat, n2 = 2+i and Z2 = 1.

Fig. 3.
Fig. 3.

P 11 for cubic particles with size parameter x = 8 and n = 1.5, Z = 1. Results for a particle with the same shape but n = 1.5, Z = 1/1.5 are also included for comparison.

Fig. 4.
Fig. 4.

(a) The projections on the x-y plane of various homogeneous impedance-matched columns, an octagonal column, a hexagonal column, a cube, and a rectangular column, oriented with one base facing the incident radiation. (b) P 11 in the ϕ = 0° plane for the above columns. The refractive index was n = 1.21. Inset shows the same results from 160° to 180°.

Fig. 5.
Fig. 5.

Scattering off coated-cubes with a regular dielectric core (Z1 = 1/n1) and an absorbing shell with unit impedance (n2 = 2+i and Z2 = 1). The size parameters are x = 4 and y = 8. Results for n1 = 1.1 and 1.5 cases are shown. The dash curves show the scattering off the dielectric cores if naked for comparison.

Fig. 6.
Fig. 6.

Electric field distribution in the vicinity of cloaked and uncloaked spheres, with scales in unit of λ. (a) A cloaked sphere with R2 = 8λ/2π and R1 = 0.3R2; (b) A cloaked sphere with R2 = 8λ/2π and R1 = 0.5R2. (c) An uncloaked homogeneous impedancematched sphere with ε = µ = 1.21 and R = 8λ/2π. Black circles indicate the inner and outer boundaries of the cloak, and the incident radiation propagates from the left to the right.

Fig. 7.
Fig. 7.

P11 for a cloaked sphere with R2 = 5λ/2π and R1 = 0.2R2. Refractive indices for the cloaked region n = 1.3, 1.5 and 1.7 were simulated. Dash lines are P11’s for corresponding uncloaked spheres.

Tables (1)

Tables Icon

Table 1. Simulated scattering efficiencies of cloaked spheres and the corresponding uncloaked cores with various refractive indices n.

Equations (25)

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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 ̂ ] ,
ε t = ε 0 R 2 R 2 R 1 and μ t = μ 0 R 2 R 2 R 1
ε r ( r ) = ε t ( r R 1 ) 2 r 2 and μ r ( r ) = μ t ( r R 1 ) 2 r 2
χ m , j ( ll ) = 3 4 π n μ j ( ll ) 1 μ j ( ll ) + 2
[ α = j 1 P j χ = m , j 1 M j ] = [ E inc , j H inc , j ] k j [ A jk ( ee ) A jk ( eh ) A jk ( he ) A jk ( hh ) ] . [ P k M k ] ,
E inc , j = E 0 exp ( i k · r j iωt )
H inc , j = H 0 exp ( i k · r j iωt )
A jk ( ee ) P k = exp ( i k r jk ) r jk 3 { k 2 r jk × ( r jk × P k ) +
( 1 ikr jk ) r jk 2 [ r jk 2 P k 3 r jk ( r jk · P k ) ] } ,
A jk ( eh ) M k = exp ( ikr jk ) r jk 3 ( r jk × M k ) ( r jk 1 ik ) ,
A jk ( he ) P k = exp ( ikr jk ) r jk 3 ( r jk × P k ) ( 1 ik r jk ) ,
A jk ( hh ) M k = exp ( i k r jk ) r jk 3 { k 2 r jk × ( r jk × M k ) +
( 1 ikr jk ) r jk 2 [ r jk 2 M k 3 r jk ( r jk · M k ) ] } ,
A jj ( ee ) = α = j 1 , A jj ( eh ) = 0 , A jj ( he ) = 0 , A jj ( hh ) = χ = m , j 1 .
k = 1 N [ A jk ( ee ) A jk ( eh ) A jk ( he ) A jk ( hh ) ] · [ P k M k ] = [ E inc , j H inc , j ] .
P 1 = ( x , y , z ) , M 1 = ( u , v , w ) .
k = 1 N [ A jk ( hh ) A jk ( he ) A jk ( eh ) A jk ( ee ) ] · [ M k P k ] = [ E inc , j H inc , j ] .
A jk (ee) = A jk (hh) , A jk (eh) = A jk (he) .
k = 1 N [ A jk ( ee ) A jk ( eh ) A jk ( he ) A jk ( hh ) ] · [ M k P k ] = [ E inc , j H inc , j ] .
P = M ,   M = P ,
P 2 = ( v , u , w ) , M 2 = ( y , x , z ) .
P 3 = ( x , y , z ) , M 3 = ( u , v , w ) ,
P 4 = ( v , u , w ) , M 4 = ( y , x , z ) .
E 1234 = j = 1 4 [ P j ( n · P j ) n n × M j ] ,

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