Abstract

A slab of negatively refracting material is known to focus light and if n=-1 the focussing will be perfect, producing an image which is an exact replica of the object. Magnifying the image requires a new design concept in which the surface of the negatively refracting lens is curved. Here we show how a hollow cylinder of material can be designed to magnify an image but otherwise with the same perfection as the original lens. Curvature requires that ε and µ are now a function of position.

© 2003 Optical Society of America

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References

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  1. V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Soviet Physics USPEKHI 10, 509 (1968).
    [Crossref]
  2. R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science 292, 79 (2001).
    [Crossref]
  3. J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 853966 (2000).
    [Crossref] [PubMed]
  4. D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, “3D Wire Mesh Photonic Crystals,” Phys Rev Lett 76, 2480 (1996).
    [Crossref] [PubMed]
  5. J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett 764773 (1996)
    [Crossref] [PubMed]
  6. J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. [Condensed Matter]  10, 4785 (1998).
  7. J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques 47, 2075 (1999).
    [Crossref]
  8. D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,”  47, 2059–2074 (1999).
    [Crossref]
  9. R.F.J. Broas, D.F. Sievenpiper, and E. Yablonovitch “A high-impedance ground plane applied to a cell phone handset geometry,” IEEE Trans. Micr. Theory and Tech. 49, 1262–1265 (2001).
    [Crossref]
  10. M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
    [Crossref]
  11. Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos, and J.B. Pendry “All-Angle Negative Refraction without Negative Effective Index,” Phys. Rev. Rapid Communications B65, 201104(R) (2002).
  12. J.B. Pendry and S.A. Ramakrishna “Near Field Lenses in Two Dimensions,” J. Phys. [Condensed Matter]  141–17 (2002).
  13. A.J. Ward and J.B. Pendry “Refraction and Geometry in Maxwell’s Equations,” Journal of Modern Optics 43773–793 (1996).
    [Crossref]
  14. R.H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106, 874 (1957).
    [Crossref]

2002 (2)

Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos, and J.B. Pendry “All-Angle Negative Refraction without Negative Effective Index,” Phys. Rev. Rapid Communications B65, 201104(R) (2002).

J.B. Pendry and S.A. Ramakrishna “Near Field Lenses in Two Dimensions,” J. Phys. [Condensed Matter]  141–17 (2002).

2001 (3)

R.F.J. Broas, D.F. Sievenpiper, and E. Yablonovitch “A high-impedance ground plane applied to a cell phone handset geometry,” IEEE Trans. Micr. Theory and Tech. 49, 1262–1265 (2001).
[Crossref]

M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
[Crossref]

R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science 292, 79 (2001).
[Crossref]

2000 (1)

J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 853966 (2000).
[Crossref] [PubMed]

1999 (2)

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques 47, 2075 (1999).
[Crossref]

D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,”  47, 2059–2074 (1999).
[Crossref]

1998 (1)

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. [Condensed Matter]  10, 4785 (1998).

1996 (3)

D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, “3D Wire Mesh Photonic Crystals,” Phys Rev Lett 76, 2480 (1996).
[Crossref] [PubMed]

J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett 764773 (1996)
[Crossref] [PubMed]

A.J. Ward and J.B. Pendry “Refraction and Geometry in Maxwell’s Equations,” Journal of Modern Optics 43773–793 (1996).
[Crossref]

1968 (1)

V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Soviet Physics USPEKHI 10, 509 (1968).
[Crossref]

1957 (1)

R.H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106, 874 (1957).
[Crossref]

Alexopoulos, F.J.

D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,”  47, 2059–2074 (1999).
[Crossref]

Broas, R.F.J.

R.F.J. Broas, D.F. Sievenpiper, and E. Yablonovitch “A high-impedance ground plane applied to a cell phone handset geometry,” IEEE Trans. Micr. Theory and Tech. 49, 1262–1265 (2001).
[Crossref]

D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,”  47, 2059–2074 (1999).
[Crossref]

Gilderdale, D.J.

M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
[Crossref]

Hajnal, J.V.

M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
[Crossref]

Holden, A.J.

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques 47, 2075 (1999).
[Crossref]

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. [Condensed Matter]  10, 4785 (1998).

J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett 764773 (1996)
[Crossref] [PubMed]

Joannopoulos, J.D.

Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos, and J.B. Pendry “All-Angle Negative Refraction without Negative Effective Index,” Phys. Rev. Rapid Communications B65, 201104(R) (2002).

Johnson, Steven G.

Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos, and J.B. Pendry “All-Angle Negative Refraction without Negative Effective Index,” Phys. Rev. Rapid Communications B65, 201104(R) (2002).

Larkman, D.J.

M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
[Crossref]

Luo, Chiyan

Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos, and J.B. Pendry “All-Angle Negative Refraction without Negative Effective Index,” Phys. Rev. Rapid Communications B65, 201104(R) (2002).

Pendry, J.B.

Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos, and J.B. Pendry “All-Angle Negative Refraction without Negative Effective Index,” Phys. Rev. Rapid Communications B65, 201104(R) (2002).

J.B. Pendry and S.A. Ramakrishna “Near Field Lenses in Two Dimensions,” J. Phys. [Condensed Matter]  141–17 (2002).

M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
[Crossref]

J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 853966 (2000).
[Crossref] [PubMed]

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques 47, 2075 (1999).
[Crossref]

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. [Condensed Matter]  10, 4785 (1998).

J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett 764773 (1996)
[Crossref] [PubMed]

A.J. Ward and J.B. Pendry “Refraction and Geometry in Maxwell’s Equations,” Journal of Modern Optics 43773–793 (1996).
[Crossref]

Ramakrishna, S.A.

J.B. Pendry and S.A. Ramakrishna “Near Field Lenses in Two Dimensions,” J. Phys. [Condensed Matter]  141–17 (2002).

Ritchie, R.H.

R.H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106, 874 (1957).
[Crossref]

Robbins, D.J.

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques 47, 2075 (1999).
[Crossref]

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. [Condensed Matter]  10, 4785 (1998).

Schultz, S.

R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science 292, 79 (2001).
[Crossref]

Shelby, R.A.

R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science 292, 79 (2001).
[Crossref]

Sickmiller, M.E.

D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, “3D Wire Mesh Photonic Crystals,” Phys Rev Lett 76, 2480 (1996).
[Crossref] [PubMed]

Sievenpiper, D.F.

R.F.J. Broas, D.F. Sievenpiper, and E. Yablonovitch “A high-impedance ground plane applied to a cell phone handset geometry,” IEEE Trans. Micr. Theory and Tech. 49, 1262–1265 (2001).
[Crossref]

D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,”  47, 2059–2074 (1999).
[Crossref]

D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, “3D Wire Mesh Photonic Crystals,” Phys Rev Lett 76, 2480 (1996).
[Crossref] [PubMed]

Smith, D.R.

R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science 292, 79 (2001).
[Crossref]

Stewart, W.J.

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques 47, 2075 (1999).
[Crossref]

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. [Condensed Matter]  10, 4785 (1998).

J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett 764773 (1996)
[Crossref] [PubMed]

Veselago, V.G.

V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Soviet Physics USPEKHI 10, 509 (1968).
[Crossref]

Ward, A.J.

A.J. Ward and J.B. Pendry “Refraction and Geometry in Maxwell’s Equations,” Journal of Modern Optics 43773–793 (1996).
[Crossref]

Wiltshire, M.C.K.

M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
[Crossref]

Yablonovitch, E.

R.F.J. Broas, D.F. Sievenpiper, and E. Yablonovitch “A high-impedance ground plane applied to a cell phone handset geometry,” IEEE Trans. Micr. Theory and Tech. 49, 1262–1265 (2001).
[Crossref]

D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,”  47, 2059–2074 (1999).
[Crossref]

D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, “3D Wire Mesh Photonic Crystals,” Phys Rev Lett 76, 2480 (1996).
[Crossref] [PubMed]

Young, I.R

M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
[Crossref]

Youngs, I.

J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett 764773 (1996)
[Crossref] [PubMed]

Zhang, L.

D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,”  47, 2059–2074 (1999).
[Crossref]

IEEE Trans. Micr. Theory and Tech. (2)

D.F. Sievenpiper, L. Zhang, R.F.J. Broas, F.J. Alexopoulos, and E. Yablonovitch, IEEE Trans. Micr. Theory and Tech. “High impedance electromagnetic surfaces with a forbidden frequency band,”  47, 2059–2074 (1999).
[Crossref]

R.F.J. Broas, D.F. Sievenpiper, and E. Yablonovitch “A high-impedance ground plane applied to a cell phone handset geometry,” IEEE Trans. Micr. Theory and Tech. 49, 1262–1265 (2001).
[Crossref]

IEEE Transactions on Microwave Theory and Techniques (1)

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Magnetism from Conductors and Enhanced Non-Linear Phenomena,” IEEE Transactions on Microwave Theory and Techniques 47, 2075 (1999).
[Crossref]

J. Phys. (2)

J.B. Pendry and S.A. Ramakrishna “Near Field Lenses in Two Dimensions,” J. Phys. [Condensed Matter]  141–17 (2002).

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. [Condensed Matter]  10, 4785 (1998).

Journal of Modern Optics (1)

A.J. Ward and J.B. Pendry “Refraction and Geometry in Maxwell’s Equations,” Journal of Modern Optics 43773–793 (1996).
[Crossref]

Phys Rev Lett (2)

D.F. Sievenpiper, M.E. Sickmiller, and E. Yablonovitch, “3D Wire Mesh Photonic Crystals,” Phys Rev Lett 76, 2480 (1996).
[Crossref] [PubMed]

J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys Rev Lett 764773 (1996)
[Crossref] [PubMed]

Phys. Rev. (1)

R.H. Ritchie, “Plasma Losses by Fast Electrons in Thin Films,” Phys. Rev. 106, 874 (1957).
[Crossref]

Phys. Rev. Lett. (1)

J.B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 853966 (2000).
[Crossref] [PubMed]

Phys. Rev. Rapid Communications (1)

Chiyan Luo, Steven G. Johnson, J.D. Joannopoulos, and J.B. Pendry “All-Angle Negative Refraction without Negative Effective Index,” Phys. Rev. Rapid Communications B65, 201104(R) (2002).

Science (2)

M.C.K. Wiltshire, J.B. Pendry, I.R Young, D.J. Larkman, D.J. Gilderdale, and J.V. Hajnal., “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291848–851 (2001).
[Crossref]

R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of negative index of refraction,” Science 292, 79 (2001).
[Crossref]

Soviet Physics USPEKHI (1)

V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Soviet Physics USPEKHI 10, 509 (1968).
[Crossref]

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Figures (2)

Fig. 1.
Fig. 1.

The annular lens produces a magnified image of internal objects, and a demagnified image of external objects. The magnifying factor is b 2/a 2. The lens is myopic: only objects closer than r=b 2 a can form an image inside the annulus. Conversely objects within the annulus and closer to the centre than r=a 2/b to the centre will not produce an image outside the annulus.

Fig. 2.
Fig. 2.

Other lenses with cylindrical geometry are possible. Here we see the crescent lens in which the inner and outer surfaces touch at the origin.

Equations (37)

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ε = 1 , μ = 1
z ' = ln z
ε ( r ) = μ ( r ) > 0 , r < a
ε ( r ) = μ ( r ) < 0 , a < r < b
ε ( r ) = μ ( r ) > 0 , r > b
x = r 0 e 0 cos ϕ , y = r 0 e 0 sin ϕ , z = Z
ε ˜ i = ε i Q 1 Q 2 Q 3 Q i 2 , μ ˜ i = μ i Q 1 Q 2 Q 3 Q i 2
E ˜ i = Q i E i , H ˜ i = Q i H i
Q i 2 = ( x q i ) 2 + ( y q i ) 2 + ( z q i ) 2
Q = r 0 0 e 2 0 cos 2 ϕ + e 2 0 sin 2 ϕ = r 0 0 e 0
Q ϕ = r 0 e 2 0 sin 2 ϕ + e 2 0 cos 2 ϕ = r 0 e 0
Q Z = 1
Q Q ϕ Q Z = r 0 2 0 e 2 0
ε ˜ = 0 ε , ε ˜ ϕ = 0 1 ε ϕ , ε ˜ Z = r 0 2 o e 2 0 ε z
μ ˜ = 0 μ , μ ˜ ϕ = 0 1 μ ϕ , μ ˜ Z = r 0 2 0 e 2 0 μ z
ε = μ = + 1 , ε ϕ = μ ϕ = + 1 , ε z = μ z = + r 0 2 e 2 = + r 2 , r < a ,
ε = μ = 1 , ε ϕ = μ ϕ = 1 , ε z = μ z = r 0 2 e 2 = r 2 , b > r > a ,
ε = μ = + 1 ε ϕ = μ ϕ = + 1 , ε z = μ z = + r 0 2 e 2 = + r 2 , r > b
ε ˜ = μ ˜ = + 1 , ε ˜ ϕ = μ ˜ ι = + 1 , ε ˜ Z = μ ˜ Z = + 1 , < 0 ln ( a r 0 ) ,
ε ˜ = μ ˜ = 1 , ε ˜ ϕ = μ ˜ ι = 1 , ε ˜ Z = μ ˜ Z = 1 , 0 ln ( a r 0 ) < < 0 ln ( b r 0 ) ,
ε ˜ = μ ˜ = + 1 , ε ˜ ϕ = μ ˜ ι = + 1 , ε ˜ Z = μ ˜ Z = + 1 , > 0 ln ( b r 0 )
1 = 0 ln ( r 1 r 0 ) < 0 ln ( a r 0 )
2 = 1 + 2 0 ln ( b a ) = 0 ln ( r 2 r 0 )
r 2 = r 1 b 2 a 2
z ' = z 1
x = x ' ( x ' 2 + y ' 2 ) , y = y ' ( x ' 2 + y ' 2 ) , z = z '
Q x ' = [ 1 ( x ' 2 + y ' 2 ) 2 x ' 2 ( x ' 2 + y ' 2 ) 2 ] 2 + [ 2 x ' y ' ( x ' 2 + y ' 2 ) 2 ] 2
= [ x ' 2 + y ' 2 2 x ' 2 ] 2 + 4 x ' 2 y ' 2 ( x ' 2 + y ' 2 ) 2 = 1 ( x ' 2 + y ' 2 )
Q y ' = [ 2 x ' y ' ( x ' 2 + y ' 2 ) 2 ] 2 + [ 1 ( x ' 2 + y ' 2 ) 2 y ' 2 ( x ' 2 + y ' 2 ) 2 ] 2
= 4 x ' 2 y ' 2 + [ x ' 2 + y ' 2 2 x ' 2 ] 2 ( x ' 2 + y ' 2 ) 2 = 1 ( x ' 2 + y ' 2 )
Q z ' = 1
Q x ' Q y ' Q z ' = ( x ' 2 + y ' 2 ) 2
ε ˜ x = ε x , ε ˜ y = ε y , ε ˜ z = ( x ' 2 + y ' 2 ) 2 ε z = ( x 2 + y 2 ) 2 ε z
μ ˜ x = μ x , μ ˜ y = μ y , μ ˜ z = ( x ' 2 + y ' 2 ) 2 μ z = ( x 2 + y 2 ) 2 μ z
ε x = μ x = + 1 , ε y = μ y = + 1 , ε z = μ z = + r 2 , x x 2 + y 2 > a 1 ,
ε x = μ x = 1 , ε y = μ y = 1 , ε z = μ z = r 2 , b 1 < x x 2 + y 2 < a 1 ,
ε x = μ x = + 1 , ε y = μ y = + 1 , ε z = μ z = + r 2 , x x 2 + y 2 < b 1

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