Abstract

We analyze the effectiveness of cloaking an infinite cylinder from observations by electromagnetic waves in three dimensions. We show that, as truncated approximations of the ideal permittivity and permeability material parameters tend towards the singular ideal cloaking values, the D and B fields blow up near the cloaking surface. Since the metamaterials used to implement cloaking are based on effective medium theory, the resulting large variation in D and B poses a challenge to the suitability of the field-averaged characterization of ε and μ. We also consider cloaking with and without the SHS (soft-and-hard surface) lining. We demonstrate numerically that cloaking is significantly improved by the SHS lining, with both the far field of the scattered wave significantly reduced and the blow up of D and B prevented.

© 2007 Optical Society of America

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References

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  1. A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot detected in EIT," Physiological Measurement (special issue on Impedance Tomography),  24, 413-420 (2003).
    [CrossRef] [PubMed]
  2. A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calder´on’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).
  3. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
    [CrossRef] [PubMed]
  4. J. B. Pendry, D. Schurig and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  5. J. B. Pendry, D. Schurig, D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794 (2006).
    [CrossRef] [PubMed]
  6. A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," ArXiv.org:math.AP/0611185v1,2,3 (2006); Commun. Math. Phys. 275, 749-789 (2007).
    [CrossRef]
  7. S. Cummer, B.-I. Popa, D. Schurig, D. Smith and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E2006 Sep; 74(3 Pt 2):036621.
    [CrossRef]
  8. D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr and D. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (10 Nov. 2006).
    [CrossRef] [PubMed]
  9. W. Cai, U. Chettiar, A. Kildshev and V. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics  1, 224-227 (2007).
    [CrossRef]
  10. H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," ArXiv.org:physics/0702050v1 (2007).
  11. F. Zolla, S. Guenneau, A. Nicolet and J. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007).
    [CrossRef] [PubMed]
  12. G. Milton, M. Briane and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
    [CrossRef]
  13. S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007).
    [CrossRef]
  14. G. Milton, "New metamaterials with macroscopic behavior outside that of continuum elastodynamics," ArXiv.org:070.2202v1 (2007).
  15. S. Schelkunoff and H. Friis, Antennas: Theory and Practice, (Chapman and Hall, New York, 1952) pp. 584-585.
  16. A. Moroz, "Some negative refractive index material headlines," http://www.wavescattering. com/negative.html.
  17. R. Weder, "A rigorous time-domain analysis of full-wave electromagnetic cloaking (Invisibility)," ArXiv.org:07040248v1,2,3 (2007).
  18. A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," ArXiv.org:math-ph/0703059; Phys. Rev. Lett. to appear.
    [PubMed]
  19. A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Electromagnetic wormholes via handlebody constructions," ArXiv.org:0704.0914v1, submitted (2007).
  20. M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," ArXiv.org:0706.0655v1 (2007).
  21. Z. Ruan, M. Yan, C. Neff and M. Qiu, "Confirmation of cylindrical perfect invisibility cloak using Fourier-Besse analysis," ArXiv.org:0704.1183v1 (2007).
  22. P.-S. Kildal, "Definition of artificially soft and hard surfaces for electromagnetic waves," Electron. Lett. 24, 168-170 (1988).
    [CrossRef]
  23. P.-S. Kildal, "Artificially soft-and-hard surfaces in electromagnetics," IEEE Trans Antennas Propag. 10, 1537-1544 (1990).
    [CrossRef]
  24. I. Hanninen, I. Lindell, and A. Sihvola, "Realization of generalized Soft-and-Hard Boundary," Progr. Electromagn. Res. 64, 317-333 (2006).
    [CrossRef]
  25. I. M. Gel’fand and G. E. Shilov, Generalized Functions, I-V (Academic Press, New York, 1964).
  26. A. Bossavit, A. Computational electromagnetism. Variational formulations, complementarity, edge elements, (Academic Press Inc., San Diego, CA, 1998).
  27. I. Lindell, Differential Forms in Electromagnetics, (Wiley-IEEE Press, 2004).
    [CrossRef]
  28. C. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory. Second edition. Appl. Math. Sci. (Springer-Verlag, Berlin, 1998) Vol. 93.
  29. M. Abramowitz and I. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables (U.S. Gov. Printing Office, Washington, D.C.,1964).
  30. G. Milton, The Theory of Composites (Cambridge U. Press, 2001).
  31. D. Smith and J. Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B 23, 391-403 (2006).
    [CrossRef]
  32. R. Kohn, H. Shen, M. Vogelius and M. Weinstein, "Cloaking via change of variables in electric impedance tomography," preprint, http://math.nyu.edu/faculty/kohn/papers/KSVW-cloaking.pdf> (2007).

2007 (3)

W. Cai, U. Chettiar, A. Kildshev and V. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics  1, 224-227 (2007).
[CrossRef]

F. Zolla, S. Guenneau, A. Nicolet and J. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007).
[CrossRef] [PubMed]

S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007).
[CrossRef]

2006 (6)

G. Milton, M. Briane and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
[CrossRef]

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

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," ArXiv.org:math.AP/0611185v1,2,3 (2006); Commun. Math. Phys. 275, 749-789 (2007).
[CrossRef]

S. Cummer, B.-I. Popa, D. Schurig, D. Smith and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E2006 Sep; 74(3 Pt 2):036621.
[CrossRef]

I. Hanninen, I. Lindell, and A. Sihvola, "Realization of generalized Soft-and-Hard Boundary," Progr. Electromagn. Res. 64, 317-333 (2006).
[CrossRef]

D. Smith and J. Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B 23, 391-403 (2006).
[CrossRef]

2003 (2)

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot detected in EIT," Physiological Measurement (special issue on Impedance Tomography),  24, 413-420 (2003).
[CrossRef] [PubMed]

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calder´on’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

1990 (1)

P.-S. Kildal, "Artificially soft-and-hard surfaces in electromagnetics," IEEE Trans Antennas Propag. 10, 1537-1544 (1990).
[CrossRef]

1988 (1)

P.-S. Kildal, "Definition of artificially soft and hard surfaces for electromagnetic waves," Electron. Lett. 24, 168-170 (1988).
[CrossRef]

Briane, M.

G. Milton, M. Briane and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
[CrossRef]

Cai, W.

W. Cai, U. Chettiar, A. Kildshev and V. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics  1, 224-227 (2007).
[CrossRef]

Chettiar, U.

W. Cai, U. Chettiar, A. Kildshev and V. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics  1, 224-227 (2007).
[CrossRef]

Cummer, S.

S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007).
[CrossRef]

S. Cummer, B.-I. Popa, D. Schurig, D. Smith and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E2006 Sep; 74(3 Pt 2):036621.
[CrossRef]

Greenleaf, A.

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," ArXiv.org:math.AP/0611185v1,2,3 (2006); Commun. Math. Phys. 275, 749-789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot detected in EIT," Physiological Measurement (special issue on Impedance Tomography),  24, 413-420 (2003).
[CrossRef] [PubMed]

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calder´on’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Guenneau, S.

Kildal, P.-S.

P.-S. Kildal, "Artificially soft-and-hard surfaces in electromagnetics," IEEE Trans Antennas Propag. 10, 1537-1544 (1990).
[CrossRef]

P.-S. Kildal, "Definition of artificially soft and hard surfaces for electromagnetic waves," Electron. Lett. 24, 168-170 (1988).
[CrossRef]

Kildshev, A.

W. Cai, U. Chettiar, A. Kildshev and V. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics  1, 224-227 (2007).
[CrossRef]

Kurylev, Y.

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," ArXiv.org:math.AP/0611185v1,2,3 (2006); Commun. Math. Phys. 275, 749-789 (2007).
[CrossRef]

Lassas, M.

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," ArXiv.org:math.AP/0611185v1,2,3 (2006); Commun. Math. Phys. 275, 749-789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calder´on’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot detected in EIT," Physiological Measurement (special issue on Impedance Tomography),  24, 413-420 (2003).
[CrossRef] [PubMed]

Milton, G.

G. Milton, M. Briane and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
[CrossRef]

Nicolet, A.

Pendry, J.

Pendry, J. B.

Popa, B.-I.

S. Cummer, B.-I. Popa, D. Schurig, D. Smith and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E2006 Sep; 74(3 Pt 2):036621.
[CrossRef]

Schurig, D.

S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007).
[CrossRef]

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

S. Cummer, B.-I. Popa, D. Schurig, D. Smith and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E2006 Sep; 74(3 Pt 2):036621.
[CrossRef]

Shalaev, V.

W. Cai, U. Chettiar, A. Kildshev and V. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics  1, 224-227 (2007).
[CrossRef]

Smith, D.

S. Cummer, B.-I. Popa, D. Schurig, D. Smith and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E2006 Sep; 74(3 Pt 2):036621.
[CrossRef]

D. Smith and J. Pendry, "Homogenization of metamaterials by field averaging," J. Opt. Soc. Am. B 23, 391-403 (2006).
[CrossRef]

Smith, D. R.

Uhlmann, G.

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," ArXiv.org:math.AP/0611185v1,2,3 (2006); Commun. Math. Phys. 275, 749-789 (2007).
[CrossRef]

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot detected in EIT," Physiological Measurement (special issue on Impedance Tomography),  24, 413-420 (2003).
[CrossRef] [PubMed]

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calder´on’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Willis, J.

G. Milton, M. Briane and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
[CrossRef]

Zolla, F.

Commun. Math. Phys. (1)

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," ArXiv.org:math.AP/0611185v1,2,3 (2006); Commun. Math. Phys. 275, 749-789 (2007).
[CrossRef]

Electron. Lett. (1)

P.-S. Kildal, "Definition of artificially soft and hard surfaces for electromagnetic waves," Electron. Lett. 24, 168-170 (1988).
[CrossRef]

IEEE Trans Antennas Propag. (1)

P.-S. Kildal, "Artificially soft-and-hard surfaces in electromagnetics," IEEE Trans Antennas Propag. 10, 1537-1544 (1990).
[CrossRef]

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

Math. Res. Lett. (1)

A. Greenleaf, M. Lassas and G. Uhlmann, "On nonuniqueness for Calder´on’s inverse problem," Math. Res. Lett. 10, 685-693 (2003).

Nat. Photonics (1)

W. Cai, U. Chettiar, A. Kildshev and V. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics  1, 224-227 (2007).
[CrossRef]

New J. Phys. (1)

G. Milton, M. Briane and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006).
[CrossRef]

New Jour. Physics (1)

S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

S. Cummer, B.-I. Popa, D. Schurig, D. Smith and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E2006 Sep; 74(3 Pt 2):036621.
[CrossRef]

Physiological Measurement (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot detected in EIT," Physiological Measurement (special issue on Impedance Tomography),  24, 413-420 (2003).
[CrossRef] [PubMed]

Progr. Electromagn. Res. (1)

I. Hanninen, I. Lindell, and A. Sihvola, "Realization of generalized Soft-and-Hard Boundary," Progr. Electromagn. Res. 64, 317-333 (2006).
[CrossRef]

Other (19)

I. M. Gel’fand and G. E. Shilov, Generalized Functions, I-V (Academic Press, New York, 1964).

A. Bossavit, A. Computational electromagnetism. Variational formulations, complementarity, edge elements, (Academic Press Inc., San Diego, CA, 1998).

I. Lindell, Differential Forms in Electromagnetics, (Wiley-IEEE Press, 2004).
[CrossRef]

C. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory. Second edition. Appl. Math. Sci. (Springer-Verlag, Berlin, 1998) Vol. 93.

M. Abramowitz and I. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables (U.S. Gov. Printing Office, Washington, D.C.,1964).

G. Milton, The Theory of Composites (Cambridge U. Press, 2001).

R. Kohn, H. Shen, M. Vogelius and M. Weinstein, "Cloaking via change of variables in electric impedance tomography," preprint, http://math.nyu.edu/faculty/kohn/papers/KSVW-cloaking.pdf> (2007).

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (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. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr and D. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (10 Nov. 2006).
[CrossRef] [PubMed]

H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," ArXiv.org:physics/0702050v1 (2007).

G. Milton, "New metamaterials with macroscopic behavior outside that of continuum elastodynamics," ArXiv.org:070.2202v1 (2007).

S. Schelkunoff and H. Friis, Antennas: Theory and Practice, (Chapman and Hall, New York, 1952) pp. 584-585.

A. Moroz, "Some negative refractive index material headlines," http://www.wavescattering. com/negative.html.

R. Weder, "A rigorous time-domain analysis of full-wave electromagnetic cloaking (Invisibility)," ArXiv.org:07040248v1,2,3 (2007).

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Electromagnetic wormholes and virtual magnetic monopoles from metamaterials," ArXiv.org:math-ph/0703059; Phys. Rev. Lett. to appear.
[PubMed]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, "Electromagnetic wormholes via handlebody constructions," ArXiv.org:0704.0914v1, submitted (2007).

M. Yan, Z. Ruan, and M. Qiu, "Cylindrical invisibility cloak with simplified material parameters is inherently visible," ArXiv.org:0706.0655v1 (2007).

Z. Ruan, M. Yan, C. Neff and M. Qiu, "Confirmation of cylindrical perfect invisibility cloak using Fourier-Besse analysis," ArXiv.org:0704.1183v1 (2007).

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Figures (5)

Fig. 1.
Fig. 1.

Diagram of how the map FR sends, in the plane z = 0, the components MR j of MR to the components NR j of the approximate cloaking device NR . Note that NR is the union of N 0NR 1 and NR 2 and thus is the space ℝ3, while MR is a union of components N 0NR 1 and MR 2 with boundaries identified and should not be thought of as lying in ℝ3.

Fig. 2.
Fig. 2.

The real part of the y-component of the total B-field on the line {(x,0,0) : x∈ [0,3]}. Blue solid curve is the field with no physical lining at {r = R}. Red dashed curve is the field with SHS lining on {r = R}. In the left figure, R = 1.05 and the maximal anisotropy ratio is LR = 1600. In the right figure, R = 1.01 and the maximal anisotropy ratio is LR = 40,000.

Fig. 3.
Fig. 3.

The real part of the y-component of the scattered B-field on the line {(x, 0,0): x∈ [0,3]}. Blue solid curve is the field with no physical lining at {r = R}. Red dashed line is the field with Soft-and-Hard lining on {r = R}. In the left figure, R = 1.05 and the maximal anisotropy ratio is LR = 1600. In the right figure, R = 1.01 and the maximal anisotropy ratio is LR = 40,000.

Fig. 4.
Fig. 4.

The magnitudes of the scattered B-fields on the exterior for LR = 1600. The decibel function 10 × log10(|Bsc |/|Bin |) is shown on a color scale. The left figure corresponds to the field in the absence of an SHS lining, while the right figure corresponds to the field with the SHS lining.

Fig. 5.
Fig. 5.

The magnitudes of the far field patterns of the scattered fields when LR = 1600. Black curve: far field pattern scattered from a perfectly conducting cylinder. Blue curve: Scattering from the invisibility coating without any physical lining. Red curve: Scattering from invisibility coated cylinder with a SHS lining.

Tables (2)

Tables Icon

Table 1. Fourier coefficients of scattered waves for R = 1.05

Tables Icon

Table 2. Fourier coefficients of scattered waves for R = 1.01

Equations (103)

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

× E = iωB ,
× H = iωD ,
D = εE ,
B = μH ,
× E ˜ = B ˜ , × H ˜ = D ˜ + J ˜ ,
D ˜ = ε ˜ E ˜ , B ˜ = μ ˜ H ˜ ,
max 1 j , k 3 λ j ( x ) λ k ( x ) = O ( ( r 1 ) 2 ) ,
lim r r ( E ˜ sc × e r + H ˜ sc ) = 0 ,
lim r 1 + e θ ( x ) E ˜ ( x ) = 0 , lim r 1 + e θ ( x ) H ˜ ( x ) = 0 ,
lim r 1 + e z ( x ) E ˜ ( x ) b e ( x x ) = 0 ,
lim r 1 + e z ( x ) H ˜ ( x ) b h ( x x ) = 0
( ν × E ˜ ) | Σ + ( ν × E ˜ ) | = ν × E ˜ | + = b e ( x ) e θ ,
( ν × H ˜ ) | Σ + ( ν × H ˜ ) | = ν × H ˜ | + = b h ( x ) e θ .
× E ˜ = B ˜ + K ˜ surf , × H ˜ = D ˜ + J ˜ surf .
δ ( f ) = 3 f ( x ) δ dx := f ( x ) dS ( x ) ,
L R := sup x 3 \ N 2 R ( max λ j ( x ) λ k ( x ) ) = O ( ( R 1 ) 2 ) ,
N 0 = { r 2 } ,
N 1 R = { R < r < 2 } , and
N 2 R = { r R } .
ε jk = ε 0 det ( g jk ) 1 2 g jk , μ jk = μ 0 det ( g jk ) 1 2 g jk , [ g jk ] = [ g jk ] 1 .
M 0 = { r 2 } ,
M 1 R = { ρ < r < 2 } ,
M 2 R = { r R } ,
g = [ g jk ] j , k = 1 3 = ( 1 0 0 0 r 2 0 0 0 1 ) , ε = ε 0 ( r 0 0 0 r 1 0 0 0 r ) , μ = μ 0 ( r 0 0 0 r 1 0 0 0 r ) .
F R : M 0 N 0 , F R | M 0 = id ,
F R : M 1 R N 1 R , F R | M 1 R ( r , θ , z ) = ( r 2 + 1 , θ , z ) ,
F R : M 2 R N 2 R , F R | M 2 R = id ,
( g ˜ R ) jk ( y ) = p , q = 1 3 y j x p y k x q g pq ( x ) , y = F R ( x ) ,
g ˜ = ( 4 0 0 0 4 ( r 1 ) 2 0 0 0 1 ) ,
ε ˜ = ε 0 ( ( r 1 ) 0 0 0 ( r 1 ) 1 0 0 0 4 ( r 1 ) ) , μ ˜ = μ 0 ( ( r 1 ) 0 0 0 ( r 1 ) 1 0 0 0 4 ( r 1 ) ) .
E ˜ = ( E ˜ 1 , E ˜ 2 , E ˜ 3 ) = ( E ˜ r , E ˜ θ , E ˜ z ) ,
E ˜ r = E ˜ 1 cos ( θ ) + E ˜ 2 sin ( θ ) , E ˜ θ = r ( E ˜ 1 sin ( θ ) + E ˜ 2 cos ( θ ) ) , E ˜ z = E ˜ 3 .
E ˜ 1 = E ˜ 2 = E ˜ r = E ˜ θ = 0 , E ˜ 3 ( x ) = E ˜ z ( r , θ ) .
H ˜ = 1 μ ˜ 1 ( × E ˜ ) = 1 μ ˜ 1 ( u × e z ) .
( Δ g ˜ + k 2 ) u = 0 on 3
E ˜ in ( r , θ , z ) = Ae ikr cos θ e z = A ( J 0 ( kr ) + n = 1 2 i n J n ( kr ) cos ( ) ) e z ,
H ˜ in ( r , θ , z ) = c ik μ 0 1 × E ˜ in ,
u ˜ in = A ( J 0 ( kr ) + n = 1 2 i n J n ( kr ) cos ( ) ) .
× E ˜ = B ˜ , × H ˜ = D ˜ ,
D ˜ = ε ˜ E ˜ , B ˜ = μ ˜ H ˜
E ˜ sc ( r . θ , z ) = ( n = 0 c n H n ( 1 ) ( kr ) cos ( ) ) e z ,
H ˜ sc ( r , θ , z ) = c ik μ 0 1 ( × E ˜ sc ) ,
u ˜ sc = n = 0 c n H n ( 1 ) ( kr ) cos ( ) .
E in = F * E ˜ in , H in = F * H ˜ in ,
E sc = F * E ˜ sc , H sc = F * H ˜ sc ,
E = F * E ˜ , H = F * H ˜ .
× E = iωB , × H = iωD ,
D = ε 0 E , B = μ 0 H .
E ( r , θ , z ) = ( AJ 0 ( kr ) + c 0 H 0 ( 1 ) ( kr ) + n = 0 ( 2 i n AJ n ( kr ) + c n H n ( 1 ) ( kr ) cos ( ) ) ) e z ,
H ( r , θ , z ) = c ik μ 0 1 ( × E ) ,
u ( r , θ ) = E 3 ( r , θ ) = A J 0 ( kr ) + c 0 H 0 ( 1 ) ( kr ) + n = 0 ( 2 i n A J n ( kr ) + c n H n ( 1 ) ( kr ) ) cos ( ) .
E ˜ r ( r , θ , z ) = 2 E r ( 2 ( r 1 ) , θ , z ) , E ˜ θ ( r , θ , z ) = E θ ( 2 ( r 1 ) , θ , z ) ,
E ˜ z ( r , θ , z ) = E z ( 2 ( r 1 ) , θ , z ) .
E ˜ ( r , θ , z ) = ( n = 0 a n J n ( kr ) cos ( ) ) e z ,
u ˜ ( r , θ ) = n = 0 a n J n ( kr ) cos ( ) ,
H ˜ ( r , θ , z ) = c ik μ 0 1 ( × E ˜ ) .
E ˜ θ | R + = E ˜ θ | R , E ˜ z | R + = E ˜ z | R ;
H ˜ θ | R + = H ˜ θ | R , H ˜ z | R + = H ˜ z | R ;
D ˜ r | R + = D ˜ r | R ; B ˜ r | R + = B ˜ r | R ,
u ˜ | R + = u ˜ | R ,
( R 1 ) r u ˜ | R + = R r u ˜ | R .
u + | r = ρ + ( θ , z ) = u | r = R ( θ , z ) ,
ρ r u + | r = ρ + ( θ , z ) = R r u | r = R ( θ , z ) ,
a 0 J 0 ( kR ) = A J 0 ( ) + c 0 H 0 ( 1 ) ( ) ,
a 0 Rk ( J 0 ) ( kR ) = Aρk ( J 0 ) ( ) + c 0 ρk ( H 0 ( 1 ) ) ( )
c 0 ( R ) = A ρ ( J 0 ) ( ) J 0 ( kR ) R J 0 ( ) ( J 0 ) ( kR ) ρ ( H 0 ( 1 ) ) ( ) J 0 ( kR ) R H 0 ( 1 ) ( ) ( J 0 ) ( kR ) = iA π 1 log ( ) ( 1 + o ( 1 ) ) ,
a 0 ( R ) = A ρ J 0 ( ) ( H 0 ( 1 ) ) ( ) ρ ( J 0 ) ( ) H 0 ( 1 ) ( ) ρ ( H 0 ( 1 ) ) ( ) J 0 ( kR ) R H 0 ( 1 ) ( ) ( J 0 ) ( kR ) = 2 A π ( J 0 ) ( k ) log ( ) ( 1 + o ( 1 ) ) ,
a n J n ( kR ) = A J n ( ) + c n H n ( 1 ) ( ) ,
a n Rk ( J n ) ( kR ) = Aρk ( J n ) ( ) + c n ρk ( H n ( 1 ) ) ( ) .
c n ( R ) = A ρ ( J n ) ( ) J n ( kR ) R J n ( ) ( J n ) ( kR ) ρ ( H n ( 1 ) ) ( ) J n ( kR ) R H n ( 1 ) ( ) ( J n ) ( kR ) = O ( ρ 2 n ) ,
a n ( R ) = A ρ J n ( ) ( H n ( 1 ) ) ( ) ρ ( J n ) ( ) H n ( 1 ) ( ) ρ ( H n ( 1 ) ) ( ) J n ( kR ) R H n ( 1 ) ( ) ( J n ) ( kR ) = O ( ρ n ) .
E ˜ ( r , θ , z ) = n = 0 E ˜ n ( r , θ , z ) , where E ˜ n ( r , θ , z ) = f n ( kr ) cos ( ) e z ;
H ˜ ( r , θ , z ) = n = 0 H ˜ n ( r , θ , z ) , where H ˜ n ( r , θ , z ) = c ik μ ˜ 1 ( × E ˜ n ( r , θ , z ) ) ,
E in , z 0 ( y ) = A J 0 ( kr ) = O ( 1 ) ,
E sc , z 0 ( y ) = c 0 ( R ) H 0 ( 1 ) ( kr ) = A ln ( kr ) ln ( ) ( 1 + o ( 1 ) )
H in , θ 0 ( y ) = iAc μ 0 r ( J 0 ) ( kr ) = O ( ( r ) 2 ) ;
H sc , θ 0 ( y ) = iAc μ 0 r c 0 ( R ) ( H 0 ( 1 ) ) ( kr ) = iAc μ 0 k ln ( ) ( 1 + o ( 1 ) ) .
E z 0 ( y ) = a 0 ( R ) J 0 ( kr ) = O ( 1 ) ln ( ) ;
H θ 0 ( y ) = c μ 0 i a 0 ( R ) r ( J 0 ) ( kr ) = O ( 1 ) ln ( ) .
B ˜ in , θ 0 ( r , θ ) = μ ˜ H ˜ in , θ 0 ( r , θ ) = μ ˜ H in , θ 0 ( 2 ( r 1 ) , θ ) = O ( r 1 ) ,
B ˜ sc , θ 0 ( r , θ ) = μ ˜ H ˜ sc , θ 0 ( r , θ ) = μ ˜ H sc , θ 0 ( 2 ( r 1 ) , θ ) = Aci k ( r 1 ) ln ( ) ( 1 + o ( 1 ) ) .
B ˜ θ 0 ( r , θ ) = O ( r ) ln ( ) ,
1 κ 1 + κ B ˜ θ 0 ( r , θ ) dr = Aci k ρ 2 κ 1 ( log ρ ) t dt + o ( 1 ) Aci k when R = ρ 2 + 1 1 + .
lim R 1 + B ˜ θ 0 = Aci k δ + B ˜ b , θ 0 ,
D ˜ in , z 0 ( r , θ ) = ε ˜ E ˜ in , z 0 ( r , θ ) = ε 0 ( r 1 ) E in , z 0 ( 2 ( r 1 ) , θ ) = O ( r 1 ) ;
D ˜ sc , z 0 ( r , θ ) = ε ˜ E ˜ sc , z 0 ( r , θ ) = ε 0 ( r 1 ) E sc , z 0 ( 2 ( r 1 ) , θ ) = O ( ( r 1 ) ln ( r 1 ) ) ln ( ) ,
D ˜ z 0 ( r , θ ) = ε ˜ E ˜ z 0 ( r , θ ) = ε 0 E z 0 ( r , θ ) = O ( 1 ) ln ( ) .
E ˜ in , z n = O ( ( r 1 ) n ) , E ˜ sc , z n = O ( ρ 2 n ( r 1 ) n ) ;
H ˜ in , r n = O ( ( r 1 ) n 1 ) , H ˜ sc , r n = O ( ρ 2 n ( r 1 ) n + 1 ) ,
H ˜ in , θ n = O ( ( r 1 ) n ) , H ˜ sc , θ n = O ( ρ 2 n ( r 1 ) n ) ;
D ˜ in , z n = O ( ( r 1 ) n + 1 ) , D ˜ sc , z n = O ( ρ 2 n ( r 1 ) n 1 ) ;
B ˜ in , r n = O ( ( r 1 ) n ) , B ˜ sc , r n = O ( ρ 2 n ( r 1 ) n ) ,
B ˜ in , θ n = O ( ( r 1 ) n 1 ) , B ˜ sc , θ n = O ( ρ 2 n ( r 1 ) n + 1 ) .
E ˜ z n = O ( ρ n ) ; D ˜ z n = O ( ρ n ) ;
H ˜ r n = O ( ρ n ) , H ˜ θ n = O ( ρ n ) ;
B ˜ r n = O ( ρ n ) , B ˜ θ n = O ( ρ n ) .
n = 1 E ˜ sc n , n = 1 H ˜ sc n , n = 1 B ˜ sc n , n = 1 D ˜ sc n ,
n = 1 E ˜ n , n = 1 H ˜ n , n = 1 B ˜ n , n = 1 D ˜ n ,
lim R 1 + E ˜ R = E ˜ b , lim R 1 + H ˜ R = H ˜ b ,
lim R 1 + D ˜ R = D ˜ b 1 J ˜ surf , J ˜ surf = 0 ,
lim R 1 + B ˜ R = B ˜ b + 1 K ˜ surf , K ˜ surf = A e θ δ .
× E ˜ lim = B ˜ lim + K ˜ surf , × H ˜ lim = D ˜ lim + J ˜ surf ,
D ˜ lim = ε ˜ E ˜ lim , B ˜ lim = μ ˜ H ˜ lim ,

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