Abstract

Based on the results of the operations of del, including the gradient, the curl, and the divergence obtained under nonorthogonal curvilinear coordinate transformations, we present a derivation of the scaling laws for electromagnetic waves, acoustic waves, and matter waves in a framework that is fully unified. Our derivation approach does not involve any tensor analysis or differential element analysis and may serve as an important supplement to those derivation methods established previously.

© 2011 Optical Society of America

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef] [PubMed]
  3. 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, 977–980 (2006).
    [CrossRef] [PubMed]
  4. H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).
    [CrossRef]
  5. S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (2008).
    [CrossRef]
  6. S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
    [CrossRef] [PubMed]
  7. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
    [CrossRef] [PubMed]
  8. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275, 749–789 (2007).
    [CrossRef]
  9. S. A. Cummer, M. Rahm, and D. Schurig, “Material parameters and vector scaling in transformation acoustics,” New J. Phys. 10, 115025 (2008).
    [CrossRef]
  10. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
    [CrossRef]

2009 (1)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

2008 (3)

S. A. Cummer, M. Rahm, and D. Schurig, “Material parameters and vector scaling in transformation acoustics,” New J. Phys. 10, 115025 (2008).
[CrossRef]

S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (2008).
[CrossRef]

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

2007 (2)

H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).
[CrossRef]

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

2006 (4)

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[CrossRef] [PubMed]

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

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (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, 977–980 (2006).
[CrossRef] [PubMed]

Chan, C. T.

H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).
[CrossRef]

Chen, H. Y.

H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).
[CrossRef]

Cummer, S. A.

S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (2008).
[CrossRef]

S. A. Cummer, M. Rahm, and D. Schurig, “Material parameters and vector scaling in transformation acoustics,” New J. Phys. 10, 115025 (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, 977–980 (2006).
[CrossRef] [PubMed]

Genov, D. A.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Greenleaf, A.

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

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, 977–980 (2006).
[CrossRef] [PubMed]

Kurylev, Y.

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

Lassas, M.

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

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

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

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, 977–980 (2006).
[CrossRef] [PubMed]

Pendry, J. B.

S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (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, 977–980 (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. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[CrossRef] [PubMed]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

Rahm, M.

S. A. Cummer, M. Rahm, and D. Schurig, “Material parameters and vector scaling in transformation acoustics,” New J. Phys. 10, 115025 (2008).
[CrossRef]

S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (2008).
[CrossRef]

Schurig, D.

S. A. Cummer, M. Rahm, and D. Schurig, “Material parameters and vector scaling in transformation acoustics,” New J. Phys. 10, 115025 (2008).
[CrossRef]

S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (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, 977–980 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[CrossRef] [PubMed]

Smith, D. R.

S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (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, 977–980 (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. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[CrossRef] [PubMed]

Starr, A.

S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (2008).
[CrossRef]

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, 977–980 (2006).
[CrossRef] [PubMed]

Sun, C.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Uhlmann, G.

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

Zhang, S.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Zhang, X.

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

H. Y. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).
[CrossRef]

Commun. Math. Phys. (1)

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

New J. Phys. (1)

S. A. Cummer, M. Rahm, and D. Schurig, “Material parameters and vector scaling in transformation acoustics,” New J. Phys. 10, 115025 (2008).
[CrossRef]

Opt. Express (1)

Phys. Rev. Lett. (2)

S. A. Cummer, D. R. Smith, M. Rahm, J. B. Pendry, D. Schurig, and A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell,” Phys. Rev. Lett. 100, 24301 (2008).
[CrossRef]

S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008).
[CrossRef] [PubMed]

Prog. Opt. (1)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

Science (3)

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

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (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, 977–980 (2006).
[CrossRef] [PubMed]

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Figures (1)

Fig. 1
Fig. 1

(a) Imaginary Cartesian space and (b) realistic Cartesian space . The line and the surface drawn in (a) and (b) represent the coordinate line and the coordinate surface, respectively.

Equations (31)

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

t i = x x ¯ i = t i t ^ i ,
n i = x ¯ i = n i n ^ i ,
x ¯ i x ¯ j = l x ¯ i x l x l x ¯ j = δ i j .
n i = j , k 1 2 | T | ε i j k t j × t k ,
t j × t k = i | T | ε i j k n i ,
t i = j , k 1 2 | N | ε i j k n j × n k ,
n j × n k = i | N | ε i j k t i ,
| T | | N | = 1 .
A = i t i ( n i · A ) = i n i ( t i · A ) .
f = l f x l x ^ l = l i f x ¯ i x ¯ i x l x ^ l = i f x ¯ i x ¯ i = i f x ¯ i n i ,
× A = × ( k ( t k · A ) n k ) = k ( t k · A ) × n k + k ( t k · A ) × n k = k ( t k · A ) × n k = k j x ¯ j ( t k · A ) n j × n k = k , j x ¯ j ( t k · A ) i | N | ε i j k t i = i , j , k ε i j k 1 | T | t i x ¯ j ( t k · A ) ,
· A = · ( i t i ( n i · A ) ) = · ( i j , k 1 2 | N | ε i j k n j × n k ( n i · A ) ) = i , j , k 1 2 ε i j k n j × n k · ( 1 | N | n i · A ) + i , j , k · ( 1 2 ε i j k n j × n k ) 1 | N | n i · A = i , j , k 1 2 ε i j k n j × n k · ( 1 | N | n i · A ) = i , j , k 1 2 ε i j k i | N | ε i j k t i · l x ¯ l ( 1 | N | n i · A ) n l = i , i δ i i | N | t i · l x ¯ l ( 1 | N | n i · A ) n l = i | N | x ¯ i ( 1 | N | n i · A ) ,
× E = i ω μ H ,
× H = i ω ε E ,
i , j , k ε i j k 1 | T | t i x ¯ j ( t k · E ) = i ω j μ n j ( t j · H ) .
j , k ε i j k x ¯ j ( t k · E ) = i ω j ( 1 | N | n i · μ n j ) ( t j · H ) .
¯ × E ¯ = i ω μ ¯ · H ¯ ,
E ¯ = T T E ,
H ¯ = T T H ,
μ ¯ = 1 | N | N T μ N .
¯ × H ¯ = i ω ε ¯ · E ¯ ,
ε ¯ = 1 | N | N T ε N .
· ( ρ 1 p ) + ω 2 λ p = 0 ,
i , j | N | x ¯ i ( 1 | N | n i · ρ 1 n j p x ¯ j ) + ω 2 λ p = 0 .
¯ · ( ρ ¯ 1 ¯ p ) + ω 2 λ ¯ p = 0 ,
ρ ¯ 1 = 1 | N | N T ρ 1 N ,
λ ¯ = | N | λ .
2 2 · ( m 1 φ ) + ( V E ) φ = 0 ,
2 2 ¯ · ( m ¯ 1 ¯ φ ) + ( V ¯ E ) φ = 0 ,
m ¯ 1 = 1 | N | N T m 1 N ,
V ¯ = E + 1 | N | ( V E ) .

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