Abstract

The ideal transformation optics cloaking is accompanied by shielding: external observations do not provide any indication of the presence of a cloaked object, nor is any information about the fields outside detectable inside the cloaked region. In this paper, a transformation is proposed to cloak three-dimensional objects for electromagnetic waves in sensor mode, i.e., cloaking accompanied by degraded shielding. The proposed transformation tackles the difficulty caused by the fact that the lowest multipole in three-dimensional electromagnetic radiation is dipole rather than monopole. The loss of the surface impedance of the sensor plays an important role in determining the cloaking modes: ideal cloaking, sensor cloaking and resonance.

© 2011 OSA

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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. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure. 24, 413–419 (2003).
    [CrossRef]
  3. Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
    [CrossRef]
  4. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007).
    [CrossRef]
  5. B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100, 063904 (2008).
    [CrossRef] [PubMed]
  6. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
    [CrossRef] [PubMed]
  7. H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16, 14603–14608 (2008).
    [CrossRef] [PubMed]
  8. B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
    [CrossRef] [PubMed]
  9. W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).
  14. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011).
    [CrossRef]
  15. 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 Propagat.57, 572–576 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]

2011

B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[CrossRef] [PubMed]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011).
[CrossRef]

2010

A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett. 105, 263906 (2010).
[CrossRef]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

2009

A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009).
[CrossRef] [PubMed]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).

2008

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16, 14603–14608 (2008).
[CrossRef] [PubMed]

B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

2007

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007).
[CrossRef]

2006

2003

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure. 24, 413–419 (2003).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

Alù, A.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011).
[CrossRef]

A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett. 105, 263906 (2010).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).

A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009).
[CrossRef] [PubMed]

Barbastathis, G.

B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[CrossRef] [PubMed]

Castaldi, G.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16, 14603–14608 (2008).
[CrossRef] [PubMed]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

H. Chen, X. Luo, H. Ma, and C. T. Chan, “The anti-cloak,” Opt. Express 16, 14603–14608 (2008).
[CrossRef] [PubMed]

Chen, H. S.

B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Cheng, Q.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008).
[CrossRef]

Chin, J. Y.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008).
[CrossRef]

Cui, T. J.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008).
[CrossRef]

Engheta, N.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011).
[CrossRef]

A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett. 105, 263906 (2010).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).

A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009).
[CrossRef] [PubMed]

Galdi, V.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).

Gallina, I.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011).
[CrossRef]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Cloak/anti-cloak interactions,” Opt. Express 17, 106343 (2009).

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 Propagat.57, 572–576 (2009).
[CrossRef]

Greenleaf, A.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure. 24, 413–419 (2003).
[CrossRef]

Jiang, W. X.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008).
[CrossRef]

Kong, J. A.

B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007).
[CrossRef]

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure. 24, 413–419 (2003).
[CrossRef]

Lin, X. Q.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008).
[CrossRef]

Liu, X. G.

B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[CrossRef] [PubMed]

Luo, X.

Luo, Y.

B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[CrossRef] [PubMed]

Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Ma, H.

Pendry, J. B.

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 Propagat.57, 572–576 (2009).
[CrossRef]

Ran, L. X.

Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Schurig, D.

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

Smith, D. R.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011).
[CrossRef]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure. 24, 413–419 (2003).
[CrossRef]

Wu, B. I.

B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

Yu, G. X.

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008).
[CrossRef]

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 Propagat.57, 572–576 (2009).
[CrossRef]

Zhang, B. L.

B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[CrossRef] [PubMed]

B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

Zhang, J. J.

Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Commun. Math. Phys.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007).
[CrossRef]

Nat. Mater.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev. B

Y. Luo, H. S. Chen, J. J. Zhang, L. X. Ran, and J. A. Kong, “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations,” Phys. Rev. B 77, 125127 (2008).
[CrossRef]

Phys. Rev. E

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking a sensor via transformation optics,” Phys. Rev. E 83, 016603 (2011).
[CrossRef]

Phys. Rev. Lett.

B. L. Zhang, H. S. Chen, B. I. Wu, and J. A. Kong, “Extraordinary surface voltage effect in the invisibility cloak with an active device inside,” Phys. Rev. Lett. 100, 063904 (2008).
[CrossRef] [PubMed]

B. L. Zhang, Y. Luo, X. G. Liu, and G. Barbastathis, “Macroscopic invisibility cloak for visible light,” Phys. Rev. Lett. 106, 033901 (2011).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Cloaking a sensor,” Phys. Rev. Lett. 102, 233901 (2009).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Cloaked near-field scanning optical microscope tip for noninvasive near-field imaging,” Phys. Rev. Lett. 105, 263906 (2010).
[CrossRef]

Physiol. Measure.

A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that cannot be detected by EIT,” Physiol. Measure. 24, 413–419 (2003).
[CrossRef]

Science

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

Wave Motion

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Analytical study of spherical cloak/anti-cloak interactions,” Wave Motion 48, 455–467 (2011).
[CrossRef]

Other

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 Propagat.57, 572–576 (2009).
[CrossRef]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

W. X. Jiang, T. J. Cui, G. X. Yu, X. Q. Lin, Q. Cheng, and J. Y. Chin, “Arbitrarily elliptical-cylindrical invisible cloaking,” J. Phys. D: Appl. Phys.41, 085504 (2008).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

The geometry of the sensor and the transformation function. (a) A sensor of arbitrary shape is placed inside a sphere with radius R0 and surface impedance α0, and it is able to measure the tangential electric field on the spherical surface. The cloak layer is in between spheres of radii R1 + δ and R2, where δ is an small positive number and is not shown in the figure due to its small magnitude. Other regions are free space. (b) The transformation r′ = f(r) from the physical space to the virtual space.

Fig. 2
Fig. 2

For n0 = 1, the cloaking and penetrating effects for different orders of multiples in the limit of δ approaching zero. (a) The cloaking effect. (b) Penetrating effect.

Fig. 3
Fig. 3

For n0 = 1 and δ = 10−3, the effect of surface impedance loss on the cloaking and penetrating effects. (a) The cloaking effect. (b) Penetrating effect, where the horizontal dotted line denotes the sensor mode.

Fig. 4
Fig. 4

For n0 = 1, the x component of the electric field in the xz plane for the loss tangent of (a) 10−2, (b) 10−4, and (c) 10−7, corresponding to the ideal cloaking, sensor, and resonance mode, respectively.

Fig. 5
Fig. 5

The x component of the electric field in the xz plane for the case when only the sensor exists, without the presence of the outer cloaking layer.

Fig. 6
Fig. 6

For n0 = 2, the cloaking and penetrating effects for different orders of multiples in the limit of δ approaching zero. (a) The cloaking effect. (b) Penetrating effect.

Fig. 7
Fig. 7

For n0 = 2 and δ = 10−3, the effect of surface impedance loss on the cloaking and penetrating effects. (a) The cloaking effect. (b) Penetrating effect, where the horizontal dotted line denotes the sensor mode.

Fig. 8
Fig. 8

For n0 = 2, the x component of the electric field in the xz plane for the loss tangent of (a) 10−4 and (b) 10−7, corresponding to the ideal cloaking and sensor mode, respectively.

Equations (38)

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

r = f ( r ) , θ = θ , ϕ = ϕ .
ɛ ¯ ¯ = ɛ r ( r ) r ^ r ^ + ɛ t ( r ) θ ^ θ ^ + ɛ t ( r ) ϕ ^ ϕ ^
μ ¯ ¯ = μ r ( r ) r ^ r ^ + μ t ( r ) θ ^ θ ^ + μ t ( r ) ϕ ^ ϕ ^ ,
ɛ r = ɛ 0 f 2 ( r ) r 2 f ( r ) , ɛ t = ɛ 0 f ( r )
μ r = μ 0 f 2 ( r ) r 2 f ( r ) , μ t = μ 0 f ( r ) .
B = × ( r ^ f Φ M ) ,
1 f 2 sin θ [ θ ( sin θ Φ M θ ) + 1 sin θ 2 Φ M ϕ 2 ] + 2 Φ M f 2 + k 0 2 Φ M = 0 ,
Φ M = n = 1 N m = n n A nm B ^ n ( k 0 f ) Y n m ( θ , ϕ ) ,
H = μ 0 1 1 r sin θ Φ M ϕ θ ^ μ 0 1 1 r Φ M θ ϕ ^
E = μ 0 1 ɛ 0 1 i ω [ f ( 2 Φ M f 2 + k 0 2 Φ M ) r ^ + 1 r 2 Φ M f θ θ ^ + 1 r sin θ 2 Φ M f ϕ ϕ ^ ] .
Φ M ext ( r , θ , ϕ ) = n m [ K nm J ^ n ( k 0 r ) + A nm H ^ n ( 1 ) ( k 0 r ) ] Y n m ( θ , ϕ ) , for r > R 2
Φ M clo ( r , θ , ϕ ) = n m [ B nm J ^ n ( k 0 f ) + C nm H ^ n ( 1 ) ( k 0 f ) ] Y n m ( θ , ϕ ) , for R 1 + δ < r < R 2
Φ M int ( r , θ , ϕ ) = n m [ D nm J ^ n ( k 0 r ) + E nm H ^ n ( 1 ) ( k 0 r ) ] Y n m ( θ , ϕ ) , for R 0 < r < R 1 + δ .
B nm J ^ n ( k 0 R 2 ) + C nm H ^ n ( 1 ) ( k 0 R 2 ) = K nm J ^ n ( k 0 R 2 ) + A nm H ^ n ( 1 ) ( k 0 R 2 )
B nm J ^ n ( k 0 R 2 ) + C nm H ^ n ( 1 ) ( k 0 R 2 ) = K nm J ^ n ( k 0 R 2 ) + A nm H ^ n ( 1 ) ( k 0 R 2 )
B nm J ^ n ( k 0 f ( R 1 + δ ) ) + C nm H ^ n ( 1 ) ( k 0 f ( R 1 + δ ) ) = D nm J ^ n ( k 0 ( R 1 + δ ) ) + E nm H ^ n ( 1 ) ( k 0 ( R 1 + δ ) )
B nm J ^ n ( k 0 f ( R 1 + δ ) ) + C nm H ^ n ( 1 ) ( k 0 f ( R 1 + δ ) ) = D nm J ^ n ( k 0 ( R 1 + δ ) ) + E nm H ^ n ( 1 ) ( k 0 ( R 1 + δ ) )
D nm J ^ n ( k 0 R 0 ) + E nm H ^ n ( 1 ) ( k 0 R 0 ) = α [ D nm J ^ n ( k 0 R 0 ) + E nm H ^ n ( 1 ) ( k 0 R 0 ) ] .
D nm = H ^ n ( 1 ) ( k 0 R 0 ) α H ^ n ( 1 ) ( k 0 R 0 ) J ^ n ( k 0 R 0 ) α J ^ n ( k 0 R 0 ) E nm g ( α ) E nm
K nm J ^ n ( k 0 f ( R 1 + δ ) ) + A nm H ^ n ( 1 ) ( k 0 f ( R 1 + δ ) ) = E nm [ g ( α ) J ^ n ( k 0 ( R 1 + δ ) ) + H ^ n ( 1 ) ( k 0 ( R 1 + δ ) ) ]
K nm J ^ n ( k 0 f ( R 1 + δ ) ) + A nm H ^ n ( 1 ) ( k 0 f ( R 1 + δ ) ) = E nm [ g ( α ) J ^ n ( k 0 ( R 1 + δ ) ) + H ^ n ( 1 ) ( k 0 ( R 1 + δ ) ) ]
E nm / K nm = i / F n ,
A nm / K nm = G n / F n ,
F n = [ g J ^ n ( k 0 ( R 1 + δ ) ) + H ^ n ( 1 ) ( k 0 ( R 1 + δ ) ) ] H ^ n ( 1 ) ( k 0 f ( R 1 + δ ) ) [ g J ^ n ( k 0 ( R 1 + δ ) ) + H ^ n ( 1 ) ( k 0 ( R 1 + δ ) ) ] H ^ n ( 1 ) ( k 0 f ( R 1 + δ ) )
G n = [ g J ^ n ( k 0 ( R 1 + δ ) ) + H ^ n ( 1 ) ( k 0 ( R 1 + δ ) ) ] J ^ n ( 1 ) ( k 0 f ( R 1 + δ ) ) + [ g J ^ n ( k 0 ( R 1 + δ ) ) + H ^ n ( 1 ) ( k 0 ( R 1 + δ ) ) ] J ^ n ( k 0 f ( R 1 + δ ) )
F n { g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) + k 0 δ [ g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) ] + ( k 0 δ ) 2 2 [ g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) ] } × ( n q n ) [ k 0 f ( R 1 + δ ) ] ( n + 1 ) { g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) + k 0 δ [ g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) ] + O ( δ 2 ) } q n [ k 0 f ( R 1 + δ ) ] n = q n [ k 0 f ( R 1 + δ ) ] ( n + 1 ) ( A 1 + A 2 + A 3 + A 4 + A 5 + A 6 ) ,
A 1 = n [ g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) ] ,
A 2 = n k 0 δ [ g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) ] ,
A 3 = k 0 f ( R 1 + δ ) [ g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) ] ,
A 4 = k 0 f ( R 1 + δ ) k 0 δ [ g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) ] ,
A 5 = n ( k 0 δ ) 2 2 [ g J ^ n ( k 0 R 1 ) + H ^ n ( 1 ) ( k 0 R 1 ) ] ,
A 6 = O ( δ 2 ) f ( R 1 + δ ) ,
E nm K nm = i q n 1 [ k 0 f ( R 1 + δ ) ] n + 1 A 1 1 .
f ( R 1 + δ ) = β δ s + o ( δ s ) ,
E nm K nm = O ( δ ( n + 1 ) s ) .
s = 1 , β = n .
α = J ^ n 0 ( k 0 R 0 ) Y ^ n 0 ( k 0 R 0 ) J ^ n 0 ( k 0 R 1 ) / Y ^ n 0 ( k 0 R 1 ) J ^ n 0 ( k 0 R 0 ) Y ^ n 0 ( k 0 R 0 ) J ^ n 0 ( k 0 R 1 ) / Y ^ n 0 ( k 0 R 1 ) .
f ( r ) = { n 0 r + n 0 R 1 for R 1 + δ r < R a h n 0 ( R a + R 1 ) R b R a ( r R a ) + n 0 ( R a + R 1 ) for R a r < R b h n 0 ( R a + R 1 ) R b R a ( r R b ) + h for R b < r R c R 2 n 0 ( R a R 1 ) R 2 R c ( r R c ) + n 0 ( R a R 1 ) for R c < r R 2 .

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