Abstract

Recent advances in metamaterial research have provided us a blueprint for realistic cloaking capabilities, and it is crucial to develop practical designs to convert concepts into real-life devices. We present two structures for optical cloaking based on high-order transformations for TM and TE polarizations respectively. These designs are possible for visible and infrared wavelengths. This critical development builds upon our previous work on nonmagnetic cloak designs and high-order transformations.

© 2008 Optical Society of America

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References

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  1. G. W. Milton, and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London, Ser. A 462, 3027-3059 (2006).
    [CrossRef]
  2. N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, "Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance," Opt. Express 15, 6314-6323 (2007).
    [CrossRef] [PubMed]
  3. A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. B 72, 016623 (2005).
    [CrossRef]
  4. M. G. Silveirinha, A. Alu, and N. Engheta, "Parallel-plate metamaterials for cloaking structures," Phys. Rev. B 75, 036603 (2007).
    [CrossRef]
  5. D. A. B. Miller, "On perfect cloaking," Opt. Express 14, 12457-12466 (2006).
    [CrossRef] [PubMed]
  6. 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] [PubMed]
  7. A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24, 413-419 (2003).
    [CrossRef] [PubMed]
  8. Y. Benveniste and T. Miloh, "Neutral inhomogeneities in conduction phenomena," J. Mech. Phys. Solids 47, 1873-1892 (1999).
    [CrossRef]
  9. A. Hendi, J. Henn, and U. Leonhardt, "Ambiguities in the scattering tomography for central potentials," Phys. Rev. Lett. 97, 073902 (2006).
    [CrossRef] [PubMed]
  10. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  11. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
    [CrossRef] [PubMed]
  12. 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]
  13. 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]
  14. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
    [CrossRef]
  15. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
    [CrossRef]
  16. R. Weder, "A rigorous analysis of high-order electromagnetic invisibility cloaks," J. Phys. A: Math. Theor. 41, 065207 (2008).
    [CrossRef]
  17. D. E. Aspnes, "Optical-Properties of Thin-Films," Thin Solid Films 89, 249-262 (1982).
    [CrossRef]
  18. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, "Imaging the near field," J. Mod. Opt. 50, 1419-1430 (2003).
  19. D. Schurig, and D. R. Smith, "Sub-diffraction imaging with compensating bilayers," New J. Phys. 7, 162 (2005).
    [CrossRef]
  20. P. A. Belov, and Y. Hao, "Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime," Phys. Rev. B 73, 113110 (2006).
    [CrossRef]
  21. S. M. Feng, and J. M. Elson, "Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms," Opt. Express 14, 216-221 (2006).
    [CrossRef] [PubMed]
  22. Z. Jacob, L. V. Alekseyev, and E. Narimanov, "Optical hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14, 8247-8256 (2006).
    [CrossRef] [PubMed]
  23. A. Salandrino, and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006).
    [CrossRef]
  24. O. Wiener, "Die Theorie des Mischkorpers fur das Feld der stationaren Stromung," Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss. 32, 509-604 (1912).
  25. D. E. Aspnes, "Bounds on Allowed Values of the Effective Dielectric Function of 2-Component Composites at Finite Frequencies," Phys. Rev. B 25, 1358-1361 (1982).
    [CrossRef]
  26. D. J. Bergman, "Exactly Solvable Microscopic Geometries and Rigorous Bounds for the Complex Dielectric-Constant of a 2-Component Composite-Material," Phys. Rev. Lett. 44, 1285-1287 (1980).
    [CrossRef]
  27. G. W. Milton, "Bounds on the Complex Dielectric-Constant of a Composite-Material," Appl. Phys. Lett. 37, 300-302 (1980).
    [CrossRef]
  28. P. B. Johnson, and R. W. Christy, "Optical-Constants of Noble-Metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  29. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1997).
  30. W. G. Spitzer, D. Kleinman, and D. Walsh, "Infrared Properties of Hexagonal Silicon Carbide," Phys. Rev. 113, 127-132 (1959).
    [CrossRef]
  31. D. Korobkin, Y. Urzhumov, and G. Shvets, "Enhanced near-field resolution in midinfrared using metamaterials," J. Opt. Soc. Am. B 23, 468-478 (2006).
    [CrossRef]
  32. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, "Near-field microscopy through a SiC superlens," Science 313, 1595-1595 (2006).
    [CrossRef] [PubMed]
  33. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, "Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles," Phys. Rev. Lett. 99, 107401 (2007).
    [CrossRef] [PubMed]
  34. S. O'Brien and J. B. Pendry, "Photonic band-gap effects and magnetic activity in dielectric composites," J. Phys. Condens. Matter. 14, 4035-4044 (2002).
    [CrossRef]
  35. K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
    [CrossRef]
  36. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, "Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies," Phys. Rev. B 72, 193103 (2005).
    [CrossRef]
  37. L. Peng, L. X. Ran, H. S. Chen, H. F. Zhang, J. A. Kong, and T. M. Grzegorczyk, "Experimental observation of left-handed behavior in an array of standard dielectric resonators," Phys. Rev. Lett. 98, 157403 (2007).
    [CrossRef] [PubMed]

2008 (1)

R. Weder, "A rigorous analysis of high-order electromagnetic invisibility cloaks," J. Phys. A: Math. Theor. 41, 065207 (2008).
[CrossRef]

2007 (6)

N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, "Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance," Opt. Express 15, 6314-6323 (2007).
[CrossRef] [PubMed]

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

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

M. G. Silveirinha, A. Alu, and N. Engheta, "Parallel-plate metamaterials for cloaking structures," Phys. Rev. B 75, 036603 (2007).
[CrossRef]

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, "Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles," Phys. Rev. Lett. 99, 107401 (2007).
[CrossRef] [PubMed]

L. Peng, L. X. Ran, H. S. Chen, H. F. Zhang, J. A. Kong, and T. M. Grzegorczyk, "Experimental observation of left-handed behavior in an array of standard dielectric resonators," Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

2006 (13)

D. Korobkin, Y. Urzhumov, and G. Shvets, "Enhanced near-field resolution in midinfrared using metamaterials," J. Opt. Soc. Am. B 23, 468-478 (2006).
[CrossRef]

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, "Near-field microscopy through a SiC superlens," Science 313, 1595-1595 (2006).
[CrossRef] [PubMed]

P. A. Belov, and Y. Hao, "Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime," Phys. Rev. B 73, 113110 (2006).
[CrossRef]

S. M. Feng, and J. M. Elson, "Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms," Opt. Express 14, 216-221 (2006).
[CrossRef] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, "Optical hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14, 8247-8256 (2006).
[CrossRef] [PubMed]

A. Salandrino, and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

D. A. B. Miller, "On perfect cloaking," Opt. Express 14, 12457-12466 (2006).
[CrossRef] [PubMed]

A. Hendi, J. Henn, and U. Leonhardt, "Ambiguities in the scattering tomography for central potentials," Phys. Rev. Lett. 97, 073902 (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. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (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]

G. W. Milton, and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London, Ser. A 462, 3027-3059 (2006).
[CrossRef]

2005 (4)

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. B 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] [PubMed]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, "Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies," Phys. Rev. B 72, 193103 (2005).
[CrossRef]

D. Schurig, and D. R. Smith, "Sub-diffraction imaging with compensating bilayers," New J. Phys. 7, 162 (2005).
[CrossRef]

2004 (1)

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
[CrossRef]

2003 (2)

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24, 413-419 (2003).
[CrossRef] [PubMed]

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, "Imaging the near field," J. Mod. Opt. 50, 1419-1430 (2003).

2002 (1)

S. O'Brien and J. B. Pendry, "Photonic band-gap effects and magnetic activity in dielectric composites," J. Phys. Condens. Matter. 14, 4035-4044 (2002).
[CrossRef]

1999 (1)

Y. Benveniste and T. Miloh, "Neutral inhomogeneities in conduction phenomena," J. Mech. Phys. Solids 47, 1873-1892 (1999).
[CrossRef]

1982 (2)

D. E. Aspnes, "Optical-Properties of Thin-Films," Thin Solid Films 89, 249-262 (1982).
[CrossRef]

D. E. Aspnes, "Bounds on Allowed Values of the Effective Dielectric Function of 2-Component Composites at Finite Frequencies," Phys. Rev. B 25, 1358-1361 (1982).
[CrossRef]

1980 (2)

D. J. Bergman, "Exactly Solvable Microscopic Geometries and Rigorous Bounds for the Complex Dielectric-Constant of a 2-Component Composite-Material," Phys. Rev. Lett. 44, 1285-1287 (1980).
[CrossRef]

G. W. Milton, "Bounds on the Complex Dielectric-Constant of a Composite-Material," Appl. Phys. Lett. 37, 300-302 (1980).
[CrossRef]

1972 (1)

P. B. Johnson, and R. W. Christy, "Optical-Constants of Noble-Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1959 (1)

W. G. Spitzer, D. Kleinman, and D. Walsh, "Infrared Properties of Hexagonal Silicon Carbide," Phys. Rev. 113, 127-132 (1959).
[CrossRef]

1912 (1)

O. Wiener, "Die Theorie des Mischkorpers fur das Feld der stationaren Stromung," Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss. 32, 509-604 (1912).

Appl. Phys. Lett. (3)

W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

G. W. Milton, "Bounds on the Complex Dielectric-Constant of a Composite-Material," Appl. Phys. Lett. 37, 300-302 (1980).
[CrossRef]

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
[CrossRef]

J. Mech. Phys. Solids (1)

Y. Benveniste and T. Miloh, "Neutral inhomogeneities in conduction phenomena," J. Mech. Phys. Solids 47, 1873-1892 (1999).
[CrossRef]

J. Mod. Opt. (1)

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, "Imaging the near field," J. Mod. Opt. 50, 1419-1430 (2003).

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

J. Phys. A: Math. Theor. (1)

R. Weder, "A rigorous analysis of high-order electromagnetic invisibility cloaks," J. Phys. A: Math. Theor. 41, 065207 (2008).
[CrossRef]

J. Phys. Condens. Matter. (1)

S. O'Brien and J. B. Pendry, "Photonic band-gap effects and magnetic activity in dielectric composites," J. Phys. Condens. Matter. 14, 4035-4044 (2002).
[CrossRef]

Nat. Photonics (1)

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

New J. Phys. (1)

D. Schurig, and D. R. Smith, "Sub-diffraction imaging with compensating bilayers," New J. Phys. 7, 162 (2005).
[CrossRef]

Opt. Express (5)

Phys. Klasse Koniglich Sachsischen Des. Wiss. (1)

O. Wiener, "Die Theorie des Mischkorpers fur das Feld der stationaren Stromung," Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss. 32, 509-604 (1912).

Phys. Rev. (1)

W. G. Spitzer, D. Kleinman, and D. Walsh, "Infrared Properties of Hexagonal Silicon Carbide," Phys. Rev. 113, 127-132 (1959).
[CrossRef]

Phys. Rev. B (7)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, "Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies," Phys. Rev. B 72, 193103 (2005).
[CrossRef]

D. E. Aspnes, "Bounds on Allowed Values of the Effective Dielectric Function of 2-Component Composites at Finite Frequencies," Phys. Rev. B 25, 1358-1361 (1982).
[CrossRef]

A. Salandrino, and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

P. B. Johnson, and R. W. Christy, "Optical-Constants of Noble-Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. B 72, 016623 (2005).
[CrossRef]

M. G. Silveirinha, A. Alu, and N. Engheta, "Parallel-plate metamaterials for cloaking structures," Phys. Rev. B 75, 036603 (2007).
[CrossRef]

P. A. Belov, and Y. Hao, "Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime," Phys. Rev. B 73, 113110 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

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] [PubMed]

A. Hendi, J. Henn, and U. Leonhardt, "Ambiguities in the scattering tomography for central potentials," Phys. Rev. Lett. 97, 073902 (2006).
[CrossRef] [PubMed]

D. J. Bergman, "Exactly Solvable Microscopic Geometries and Rigorous Bounds for the Complex Dielectric-Constant of a 2-Component Composite-Material," Phys. Rev. Lett. 44, 1285-1287 (1980).
[CrossRef]

L. Peng, L. X. Ran, H. S. Chen, H. F. Zhang, J. A. Kong, and T. M. Grzegorczyk, "Experimental observation of left-handed behavior in an array of standard dielectric resonators," Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, "Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles," Phys. Rev. Lett. 99, 107401 (2007).
[CrossRef] [PubMed]

Physiol. Meas. (1)

A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24, 413-419 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

G. W. Milton, and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London, Ser. A 462, 3027-3059 (2006).
[CrossRef]

Science (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, 977-980 (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]

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, "Near-field microscopy through a SiC superlens," Science 313, 1595-1595 (2006).
[CrossRef] [PubMed]

Thin Solid Films (1)

D. E. Aspnes, "Optical-Properties of Thin-Films," Thin Solid Films 89, 249-262 (1982).
[CrossRef]

Other (1)

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1997).

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Figures (5)

Fig. 1.
Fig. 1.

The principle of constructing a non-magnetic cloak in the TM mode with high-order transformations. The thick solid and dashed lines represent the two Wiener bounds ε (f) and ε (f), respectively. Basic material properties for this calculation: ε1 =εAg =-10.6+0.14i and ε2 =εSiO2 =2.13 at λ=532 nm.

Fig. 2.
Fig. 2.

Schematic of a cylindrical non-magnetic cloak with high-order transformations for TM polarization.

Fig. 3.
Fig. 3.

Anisotropic material parameters εr and εθ of a non-magnetic cloak made of silversilica alternating slices corresponding to the third row (λ=532 nm) in Table 1. The solid lines represent the exact parameters determined by Eq. (2), and the diamond markers show the parameters on the Wiener’s bounds given by Eq. (4).

Fig. 4.
Fig. 4.

Schematic of a cylindrical non-magnetic cloak with high-order transformations for TE polarization.

Fig. 5.
Fig. 5.

The required and the calculated effective parameters μr and εz for a cylindrical TE cloak with SiC wire arrays for λ=13.5 µm.

Tables (1)

Tables Icon

Table 1. Approximate quadratic transformations and materials for constructing a cloak with alternating slices

Equations (10)

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

ε r = μ r = ( r ' r ) g ( r ' ) r ' ; ε θ = μ θ = 1 ε r ; ε z = μ z = ( r ' r ) [ g ( r ' ) r ' ] 1
μ r = ( r ' r ) 2 [ g ( r ' ) r ' ] 2 ; μ θ = 1 ; ε z = [ g ( r ' ) r ' ] 2
ε r = ( r ' r ) 2 ; ε θ = [ g ( r ' ) r ' ] 2 ; μ z = 1
ε = f ε 1 + ( 1 f ) ε 2 ; ε = ε 1 ε 2 ( f ε 2 + ( 1 f ) ε 1 )
ε m ε d ( g ( r ' ) r ' ) 2 + ( r ' g ( r ' ) ) 2 ( ε m + ε d ) = 0
g ( 0 ) = a ; g ( b ) = b ; g ( r ' ) r ' > 0
r = g ( r ' ) = [ 1 a b + p ( r ' b ) ] r ' + a
f ( r ) = Re ( ε d ) ( g 1 ( r ) r ) 2 Re ( ε d ε m )
ε SiC = ε [ ω 2 ω L 2 + i γ ω ] [ ω 2 ω T 2 + i γ ω ]
μ r = 2 k L 1 2 L 1 J 1 ( k L 1 ) t J 1 ( k t ) + a 0 t H 1 ( 1 ) ( k t ) a 0 L 1 H 1 ( 1 ) ( k L 1 ) + c 0 t J 1 ( n k t ) n J 0 ( k L 2 2 ) a 0 H 0 ( 1 ) ( k L 2 2 )

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