Abstract

We adapt tools of transformation optics to surface plasmon polaritons (SPPs) propagating at the interface between two anisotropic media of opposite permittivity sign. We identify the role played by entries of anisotropic heterogeneous tensors of permittivity and permeability-deduced from a coordinate transformation- in the dispersion relation governing propagation of SPPs. We apply this concept to an invisibility cloak, a concentrator and a rotator for SPPs.

© 2010 Optical Society of America

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  1. S. A. Ramakrishna, and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press, 2008).
    [CrossRef]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  3. U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777 (2006).
    [CrossRef] [PubMed]
  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 (2006).
    [CrossRef] [PubMed]
  5. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007).
    [CrossRef] [PubMed]
  6. A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).
  7. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Woff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667 (1998).
    [CrossRef]
  8. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
    [CrossRef] [PubMed]
  9. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
    [CrossRef]
  10. A. Alu, and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 016623 (2005).
    [CrossRef]
  11. F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling Mechanism of Light Transmission through Metallic Films,” Phys. Rev. Lett. 95, 067403 (2005).
    [CrossRef]
  12. B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246809 (2009).
    [CrossRef]
  13. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
    [CrossRef]
  14. Y. Luo, J. Zhang, J. Wu, and H. Chen, “Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator,” Phys. Rev. B 78(12), 125108 (2008).
    [CrossRef]
  15. P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” http://arxiv.org/abs/1003.1154.
  16. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational Plasmon Optics,” http://arxiv.org/abs/1003.1326.
  17. J. Renger, M. Kadic, G. Dupont, S. Acimovic, S. Guenneau, R. Quidant, and S. Enoch, “Hidden Progress: Broadband plasmonic invisibility,” http://arxiv.org/abs/1003.5476

2009 (1)

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246809 (2009).
[CrossRef]

2008 (2)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
[CrossRef]

Y. Luo, J. Zhang, J. Wu, and H. Chen, “Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator,” Phys. Rev. B 78(12), 125108 (2008).
[CrossRef]

2007 (1)

2006 (3)

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

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

2005 (2)

A. Alu, and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 016623 (2005).
[CrossRef]

F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling Mechanism of Light Transmission through Metallic Films,” Phys. Rev. Lett. 95, 067403 (2005).
[CrossRef]

2004 (1)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
[CrossRef] [PubMed]

2003 (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Woff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667 (1998).
[CrossRef]

1994 (1)

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Alu, A.

A. Alu, and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 016623 (2005).
[CrossRef]

Baumeier, B.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246809 (2009).
[CrossRef]

Blanco, L. A.

F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling Mechanism of Light Transmission through Metallic Films,” Phys. Rev. Lett. 95, 067403 (2005).
[CrossRef]

Borisov, A. G.

F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling Mechanism of Light Transmission through Metallic Films,” Phys. Rev. Lett. 95, 067403 (2005).
[CrossRef]

Chen, H.

Y. Luo, J. Zhang, J. Wu, and H. Chen, “Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator,” Phys. Rev. B 78(12), 125108 (2008).
[CrossRef]

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
[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 (2006).
[CrossRef] [PubMed]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Woff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667 (1998).
[CrossRef]

Engheta, N.

A. Alu, and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 016623 (2005).
[CrossRef]

Garcia de Abajo, F. J.

F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling Mechanism of Light Transmission through Metallic Films,” Phys. Rev. Lett. 95, 067403 (2005).
[CrossRef]

Garcia-Vidal, F. J.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Woff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667 (1998).
[CrossRef]

Gomez-Santos, G.

F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling Mechanism of Light Transmission through Metallic Films,” Phys. Rev. Lett. 95, 067403 (2005).
[CrossRef]

Greenleaf, A.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

Guenneau, S.

Justice, B. J.

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

Lassas, M.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

Leonhardt, U.

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

Leskova, T. A.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246809 (2009).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Woff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667 (1998).
[CrossRef]

Luo, Y.

Y. Luo, J. Zhang, J. Wu, and H. Chen, “Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator,” Phys. Rev. B 78(12), 125108 (2008).
[CrossRef]

Maradudin, A. A.

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246809 (2009).
[CrossRef]

Martin-Moreno, L.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
[CrossRef] [PubMed]

McPhedran, R. C.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Milton, G. W.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Mock, J. J.

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

Nicolet, A.

Nicorovici, N. A.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Pendry, J. B.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
[CrossRef]

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

D. Schurig, J. 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 (2006).
[CrossRef] [PubMed]

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

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
[CrossRef] [PubMed]

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
[CrossRef]

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
[CrossRef]

Schurig, D.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
[CrossRef]

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

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

Shabanov, S. V.

F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling Mechanism of Light Transmission through Metallic Films,” Phys. Rev. Lett. 95, 067403 (2005).
[CrossRef]

Smith, D. R.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
[CrossRef]

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

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

Starr, A. F.

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

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Woff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667 (1998).
[CrossRef]

Uhlmann, G.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

Woff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Woff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667 (1998).
[CrossRef]

Wu, J.

Y. Luo, J. Zhang, J. Wu, and H. Chen, “Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator,” Phys. Rev. B 78(12), 125108 (2008).
[CrossRef]

Zhang, J.

Y. Luo, J. Zhang, J. Wu, and H. Chen, “Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator,” Phys. Rev. B 78(12), 125108 (2008).
[CrossRef]

Zolla, F.

Math. Res. Lett. (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderons inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Woff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667 (1998).
[CrossRef]

Opt. Lett. (1)

Photon. Nanostruct. Fundam Appl. (1)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon. Nanostruct. Fundam Appl. 6(1), 87–95 (2008).
[CrossRef]

Phys. Rev. B (2)

Y. Luo, J. Zhang, J. Wu, and H. Chen, “Interaction of an electromagnetic wave with a cone-shaped invisibility cloak and polarization rotator,” Phys. Rev. B 78(12), 125108 (2008).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. Alu, and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 016623 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling Mechanism of Light Transmission through Metallic Films,” Phys. Rev. Lett. 95, 067403 (2005).
[CrossRef]

B. Baumeier, T. A. Leskova, and A. A. Maradudin, “Cloaking from surface plasmon polaritons by a circular array of point scatterers,” Phys. Rev. Lett. 103, 246809 (2009).
[CrossRef]

Science (4)

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

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

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
[CrossRef] [PubMed]

Other (4)

P. A. Huidobro, M. L. Nesterov, L. Martin-Moreno, and F. J. García-Vidal, “Transformation Optics for Plasmonics,” http://arxiv.org/abs/1003.1154.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational Plasmon Optics,” http://arxiv.org/abs/1003.1326.

J. Renger, M. Kadic, G. Dupont, S. Acimovic, S. Guenneau, R. Quidant, and S. Enoch, “Hidden Progress: Broadband plasmonic invisibility,” http://arxiv.org/abs/1003.5476

S. A. Ramakrishna, and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press, 2008).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a) A cylindrical region filled with air above the z = 0 plane and with metal below is mapped onto a ring filled with metamaterial; The upper panel is a view from above and the lower panel is a side view; (b) Lower panel: Three-dimensional representation of the SPP propagation incident from the left on a cylindrical invisibility cloak with permittivity and permeability tensors given by Eq. (11); Upper panel: Regions involved in the geometric transformations for a cloak, a rotator and a concentrator (the latter further requiring a virtual disc of radius R 2).

Fig. 2.
Fig. 2.

(a) Reflexion of a SPP on a metallic cylinder; (b) SPP incident from the left on a cylindrical invisibility cloak surrounding the metallic cylinder; The upper panel is a view from above and the lower panel is a side view. White (out of color scale) regions are a genuine consequence of a sligth mismatch between the three-dimensional mesh and the plane z = 0.

Fig. 3.
Fig. 3.

SPP incident from the left on a concentrator (a) and a rotator (b); The upper panel is a view from above and the lower panel is a side view.

Equations (14)

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

ε = ε T 1 , and μ = μ T 1 .
× H i = 1 ω ε 0 ε i ¯ ¯ E i
H i = ( 0 , H yi , 0 ) exp { 1 ( k x x ωt ) k zi z } ,
E i = c ω H yi ( k zi ε xxi , 0 , k xi ε zzi ) exp { 1 ( k x x ωt ) k zi z }
× E i = 1 ω μ 0 μ i ¯ ¯ H i ,
k zi = ε xxi ( k x 2 ε zzi μ yyi ( ω c ) 2 ) , i = 1,2 .
k z 1 ε xx 1 + k z 2 ε xx 2 = 0 .
k x = ω c ε zz 2 ε zz 1 ( μ yy 2 ε xx 1 μ yy 1 ε xx 2 ) ε xx 1 ε zz 1 ε xx 2 ε zz 2 .
k x = ω c ε 1 ε 2 ε 1 + ε 2 .
{ r = f 1 ( r ) , θ = f 2 ( r , θ ) z = z ( 0 < r R 2 ) r = g 1 ( r ) , θ = g 2 ( r , θ ) z = z ( R 2 < r R 3 ) r = r , θ = θ z = z ( r > R 3 )
T 1 = R ( θ ) diag ( r R 2 r , r r R 2 , r R 2 α 2 r ) R T ( θ ) ,
f 1 ( r ) = α 1 r + β 1 , g 1 ( r ) = α 2 r + β 2 , and f 2 ( r , θ ) = g 2 ( r , θ ) = θ
f 1 ( r ) = r , g 1 ( r ) = r , and f 2 ( r , θ ) = θ + α 1 r + β 1 , g 2 ( r , θ ) = α 2 r + β 2 ,
T 1 = R ( θ ) ( u v w ) 1 diag ( λ 1 , λ 2 , λ 3 ) ( u v w ) R T ( θ ) ,

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