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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  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 Calderón’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).
  3. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (23 June, 2006).
    [Crossref] [PubMed]
  4. J.B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (23 June, 2006).
    [Crossref] [PubMed]
  5. J. B. Pendry, D. Schurig, and 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); Comm. 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,” Nature Photonics,  1, 224–227 (April, 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 Jour. Physics 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, 584–585).
  16. A. Moroz, “Some negative refractive index material headlines…,” http://www.wave-scattering.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-Bessel 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. Hänninen, I. Lindell, and A. Sihvola, “Realization of generalized Soft-and-Hard Boundary,” Progr. In Electro-mag. Res., PIER 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. Applied Math. Sciences, 93. (Springer-Verlag, Berlin, 1998).
  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 (4)

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); Comm. Math. Phys. 275, 749–789 (2007).
[Crossref]

W. Cai, U. Chettiar, A. Kildshev, and V. Shalaev, “Optical cloaking with metamaterials,” Nature Photonics,  1, 224–227 (April, 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 Jour. Physics 9, 45 (2007).
[Crossref]

2006 (8)

I. Hänninen, I. Lindell, and A. Sihvola, “Realization of generalized Soft-and-Hard Boundary,” Progr. In Electro-mag. Res., PIER 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]

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]

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. 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]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (23 June, 2006).
[Crossref] [PubMed]

J.B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (23 June, 2006).
[Crossref] [PubMed]

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

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 Calderón’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]

Abramowitz, M.

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

Bossavit, A.

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

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,” Nature Photonics,  1, 224–227 (April, 2007).
[Crossref]

Chan, C. T.

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

Chen, H.

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

Chettiar, U.

W. Cai, U. Chettiar, A. Kildshev, and V. Shalaev, “Optical cloaking with metamaterials,” Nature Photonics,  1, 224–227 (April, 2007).
[Crossref]

Colton, C.

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

Cummer, S.

S. Cummer and D. Schurig, “One path to acoustic cloaking,” New Jour. Physics 9, 45 (2007).
[Crossref]

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]

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]

Friis, H.

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

Gel’fand, I. M.

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

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); Comm. 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 Calderón’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

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).

Guenneau, S.

Hänninen, I.

I. Hänninen, I. Lindell, and A. Sihvola, “Realization of generalized Soft-and-Hard Boundary,” Progr. In Electro-mag. Res., PIER 64, 317–333 (2006).
[Crossref]

Justice, B.

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]

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,” Nature Photonics,  1, 224–227 (April, 2007).
[Crossref]

Kohn, R.

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).

Kress, R.

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

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); Comm. Math. Phys. 275, 749–789 (2007).
[Crossref]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes via handlebody constructions,” ArXiv.org:0704.0914v1, submitted (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]

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); Comm. 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 Calderón’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

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).

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (23 June, 2006).
[Crossref] [PubMed]

Lindell, I.

I. Hänninen, I. Lindell, and A. Sihvola, “Realization of generalized Soft-and-Hard Boundary,” Progr. In Electro-mag. Res., PIER 64, 317–333 (2006).
[Crossref]

I. Lindell, Differential Forms in Electromagnetics, Wiley-IEEE Press, 2004.
[Crossref]

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]

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

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

Mock, J.

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]

Moroz, A.

A. Moroz, “Some negative refractive index material headlines…,” http://www.wave-scattering.com/negative.html.

Neff, C.

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

Nicolet, A.

Pendry, J.

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]

D. Smith and J. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23 , 391–403 (2006).
[Crossref]

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]

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]

Pendry, J. B.

Pendry, J.B.

J.B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (23 June, 2006).
[Crossref] [PubMed]

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]

Qiu, M.

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

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

Ruan, Z.

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

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

Schelkunoff, S.

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

Schurig, D.

S. Cummer and D. Schurig, “One path to acoustic cloaking,” New Jour. Physics 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]

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]

J.B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (23 June, 2006).
[Crossref] [PubMed]

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

Shalaev, V.

W. Cai, U. Chettiar, A. Kildshev, and V. Shalaev, “Optical cloaking with metamaterials,” Nature Photonics,  1, 224–227 (April, 2007).
[Crossref]

Shen, H.

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).

Shilov, G. E.

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

Sihvola, A.

I. Hänninen, I. Lindell, and A. Sihvola, “Realization of generalized Soft-and-Hard Boundary,” Progr. In Electro-mag. Res., PIER 64, 317–333 (2006).
[Crossref]

Smith, D.

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]

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.

Smith, D.R.

J.B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (23 June, 2006).
[Crossref] [PubMed]

Starr, A.

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]

Stegun, I.

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

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); Comm. Math. Phys. 275, 749–789 (2007).
[Crossref]

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderón’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]

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Electromagnetic wormholes via handlebody constructions,” ArXiv.org:0704.0914v1, submitted (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]

Vogelius, M.

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).

Weder, R.

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

Weinstein, M.

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).

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]

Yan, M.

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-Bessel analysis,” ArXiv.org:0704.1183v1 (2007).

Zolla, F.

Comm. 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); Comm. 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 Calderón’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

Nature Photonics (1)

W. Cai, U. Chettiar, A. Kildshev, and V. Shalaev, “Optical cloaking with metamaterials,” Nature Photonics,  1, 224–227 (April, 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 Jour. Physics 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. In Electro-mag. Res., PIER (1)

I. Hänninen, I. Lindell, and A. Sihvola, “Realization of generalized Soft-and-Hard Boundary,” Progr. In Electro-mag. Res., PIER 64, 317–333 (2006).
[Crossref]

Science (3)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (23 June, 2006).
[Crossref] [PubMed]

J.B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (23 June, 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]

Other (16)

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, 584–585).

A. Moroz, “Some negative refractive index material headlines…,” http://www.wave-scattering.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-Bessel analysis,” ArXiv.org:0704.1183v1 (2007).

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. Applied Math. Sciences, 93. (Springer-Verlag, Berlin, 1998).

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).

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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 0 ∪ NR 1 and NR 2 and thus is the space ℝ3, while MR is a union of components N 0 ∪ NR 1 and MR 2 with boundaries identified and should not be thought of as lying in ℝ3.

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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.

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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.

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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.

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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.

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Tables (2)

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

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

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Equations (103)

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∇ × E = iωB ,
∇ × H = − iωD ,
D = εE ,
B = μH ,
∇ × E ˜ = iω B ˜ , ∇ × H ˜ = − iω 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 ˜ = iω B ˜ + K ˜ surf , ∇ × H ˜ = − iω D ˜ + J ˜ surf .
δ ∑ ( f ) = ∫ ℝ 3 f ( x ) δ ∑ dx := ∫ ∑ f ( x ) dS ( x ) ,
L R := sup x ∈ ℝ 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 iω μ ˜ − 1 ( ∇ × E ˜ ) = 1 iω μ ˜ − 1 ( ∇ u × e z ) .
( Δ g ˜ + k 2 ) u = 0 on ℝ 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 ⁡ ( nθ ) ) 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 ⁡ ( nθ ) ) .
∇ × E ˜ = iω B ˜ , ∇ × H ˜ = − iω D ˜ ,
D ˜ = ε ˜ E ˜ , B ˜ = μ ˜ H ˜
E ˜ sc ( r . θ , z ) = ( ∑ n = 0 ∞ c n H n ( 1 ) ( kr ) cos ⁡ ( nθ ) ) 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 ⁡ ( nθ ) .
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 ⁡ ( nθ ′ ) ) ) 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 ⁡ ( nθ ′ ) .
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 ⁡ ( nθ ) ) e z ,
u ˜ ( r , θ ) = ∑ n = 0 ∞ a n J n ( kr ) cos ⁡ ( nθ ) ,
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 ( kρ ) + c 0 H 0 ( 1 ) ( kρ ) ,
a 0 Rk ( J 0 ) ′ ( kR ) = Aρk ( J 0 ) ′ ( kρ ) + c 0 ρk ( H 0 ( 1 ) ) ′ ( kρ )
c 0 ( R ) = A ρ ( J 0 ) ′ ( kρ ) J 0 ( kR ) − R J 0 ( kρ ) ( J 0 ) ′ ( kR ) ρ ( H 0 ( 1 ) ) ′ ( kρ ) J 0 ( kR ) − R H 0 ( 1 ) ( kρ ) ( J 0 ) ′ ( kR ) = iA π 1 log ⁡ ( kρ ) ( 1 + o ( 1 ) ) ,
a 0 ( R ) = A ρ J 0 ( kρ ) ( H 0 ( 1 ) ) ′ ( kρ ) − ρ ( J 0 ) ′ ( kρ ) H 0 ( 1 ) ( kρ ) ρ ( H 0 ( 1 ) ) ′ ( kρ ) J 0 ( kR ) − R H 0 ( 1 ) ( kρ ) ( J 0 ) ′ ( kR ) = − 2 A π ( J 0 ) ′ ( k ) log ⁡ ( kρ ) ( 1 + o ( 1 ) ) ,
a n J n ( kR ) = A J n ( kρ ) + c n H n ( 1 ) ( kρ ) ,
a n Rk ( J n ) ′ ( kR ) = Aρk ( J n ) ′ ( kρ ) + c n ρk ( H n ( 1 ) ) ′ ( kρ ) .
c n ( R ) = A ρ ( J n ) ′ ( kρ ) J n ( kR ) − R J n ( kρ ) ( J n ) ′ ( kR ) ρ ( H n ( 1 ) ) ′ ( kρ ) J n ( kR ) − R H n ( 1 ) ( kρ ) ( J n ) ′ ( kR ) = O ( ρ 2 n ) ,
a n ( R ) = A ρ J n ( kρ ) ( H n ( 1 ) ) ′ ( kρ ) − ρ ( J n ) ′ ( kρ ) H n ( 1 ) ( kρ ) ρ ( H n ( 1 ) ) ′ ( kρ ) J n ( kR ) − R H n ( 1 ) ( kρ ) ( 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 ⁡ ( nθ ) 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 ⁡ ( kρ ) ( 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 ⁡ ( kρ ) ( 1 + o ( 1 ) ) .
E z 0 ( y ) = a 0 ( R ) J 0 ( kr ′ ) = O ( 1 ) ln ⁡ ( kρ ) ;
H θ 0 ( y ) = c μ 0 i a 0 ( R ) r ′ ( J 0 ) ′ ( kr ′ ) = O ( 1 ) ln ⁡ ( kρ ) .
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 ⁡ ( kρ ) ( 1 + o ( 1 ) ) .
B ˜ θ 0 ( r , θ ) = O ( r ) ln ⁡ ( kρ ) ,
∫ 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 ⁡ ( kρ ) ,
D ˜ z 0 ( r , θ ) = ε ˜ E ˜ z 0 ( r , θ ) = ε 0 E z 0 ( r , θ ) = O ( 1 ) ln ⁡ ( kρ ) .
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 iω J ˜ surf , J ˜ surf = 0 ,
lim R → 1 + B ˜ R = B ˜ b + 1 iω K ˜ surf , K ˜ surf = − A e θ δ ∑ .
∇ × E ˜ lim = iω B ˜ lim + K ˜ surf , ∇ × H ˜ lim = − iω D ˜ lim + J ˜ surf ,
D ˜ lim = ε ˜ E ˜ lim , B ˜ lim = μ ˜ H ˜ lim ,

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