Abstract

The interaction of ideal and nonideal metamaterial cylindrical cloaks with an electron beam has been studied. The exact solution for energy loss suffered by a fast electron moving inside or outside a cylindrical cloak has been established within a fully relativistic approach. The effect of various imperfect parameters on the efficiency of the cloak has been discussed. It is shown that radiation can be shielded very well by such cloaks when an electron moves either inside or outside of them. The efficiency of nonideal cloaks and the effect of various nonideal parameters on cloak invisibility can be exhibited in the spectra of electron energy loss and radiation emission. This means that the property of a cylindrical cloak can be explored by means of scanning transmission electron microscopy.

© 2009 Optical Society of America

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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. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  4. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
    [CrossRef]
  5. 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]
  6. D. A. B. Miller, "On perfect cloaking," Opt. Express 14, 12457-12466 (2006).
    [CrossRef] [PubMed]
  7. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
    [CrossRef]
  8. H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007).
    [CrossRef]
  9. H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
    [CrossRef] [PubMed]
  10. Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
    [CrossRef] [PubMed]
  11. B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
    [CrossRef]
  12. J. Xu, Y. Dong, and X. Zhang, "Electromagnetic interactions between a fast electron beam and metamaterial cloaks," Phys. Rev. E 78, 046601 (2008).
    [CrossRef]
  13. A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005).
    [CrossRef]
  14. G. W. Milton and N.-A. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London Ser. A 462, 3027-3059 (2006).
    [CrossRef]
  15. P. D. Nellist, and S. J. Pennycook, "Subangstrom resolution by underfocused incoherent transmission electron microscopy," Phys. Rev. Lett. 81, 4156-4159 (1998).
    [CrossRef]
  16. R. H. Ritchie, "Plasma losses by fast electrons in thin films," Phys. Rev. 106, 874-881 (1957).
    [CrossRef]
  17. P. M. Echenique, and J. B. Pendry, "Absorption profile at surfaces," J. Phys. C 8, 2936-2942 (1975).
    [CrossRef]
  18. R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, "Excitation of edge modes in the interaction of electron beams with dielectric wedges," Phys. Rev. B 31, 121-126 (1985).
    [CrossRef]
  19. T. L. Ferrell, and P. M. Echenique, "Generation of surface excitations on dielectric spheres by an external electron beam," Phys. Rev. Lett. 55, 1526-1529 (1985).
    [CrossRef] [PubMed]
  20. A. Rivacoba, N. Zabala, and P. M. Echenique, "Theory of energy loss in scanning transmission electron microscopy of supported small particles," Phys. Rev. Lett. 69, 3362-3365 (1992).
    [CrossRef] [PubMed]
  21. J. B. Pendry and L. Martín-Moreno, "Energy-loss by charged-particles in complex media," Phys. Rev. B 50, 5062-5073 (1994).
    [CrossRef]
  22. N. Zabala, A. Rivacoba, and P. M. Echenique, "Coupling effects in the excitations by an external electron beam near close particles," Phys. Rev. B 56, 7623-7635 (1997).
    [CrossRef]
  23. F. J. Garcia de Abajo and A. Howie, "Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics," Phys. Rev. Lett. 80, 5180-5183 (1998).
    [CrossRef]
  24. F. J. Garcia de Abajo, "Interaction of radiation and fast electrons with clusters of dielectrics: a multiple scattering approach," Phys. Rev. Lett. 82, 2776-2779 (1999).
    [CrossRef]
  25. F. J. Garcia de Abajo, "Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam," Phys. Rev. B 59, 3095-3107 (1999).
    [CrossRef]
  26. F. J. Garcia de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
    [CrossRef]
  27. T. Ochiai, and K. Ohtaka, "Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals. I. Formalism and surface plasmon polariton," Phys. Rev. B 69, 125106 (2004).
    [CrossRef]
  28. J. Xu and X. Zhang, "Relativistic energy loss and induced photon emission in the interaction of a left-handed sphere with an external electron beam," Phys. Lett. A 372, 1129-1134 (2008).
    [CrossRef]

2008

J. Xu, Y. Dong, and X. Zhang, "Electromagnetic interactions between a fast electron beam and metamaterial cloaks," Phys. Rev. E 78, 046601 (2008).
[CrossRef]

J. Xu and X. Zhang, "Relativistic energy loss and induced photon emission in the interaction of a left-handed sphere with an external electron beam," Phys. Lett. A 372, 1129-1134 (2008).
[CrossRef]

2007

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

H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007).
[CrossRef]

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

2006

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

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]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

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

2005

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

2004

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

T. Ochiai, and K. Ohtaka, "Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals. I. Formalism and surface plasmon polariton," Phys. Rev. B 69, 125106 (2004).
[CrossRef]

2003

F. J. Garcia de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

1999

F. J. Garcia de Abajo, "Interaction of radiation and fast electrons with clusters of dielectrics: a multiple scattering approach," Phys. Rev. Lett. 82, 2776-2779 (1999).
[CrossRef]

F. J. Garcia de Abajo, "Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam," Phys. Rev. B 59, 3095-3107 (1999).
[CrossRef]

1998

P. D. Nellist, and S. J. Pennycook, "Subangstrom resolution by underfocused incoherent transmission electron microscopy," Phys. Rev. Lett. 81, 4156-4159 (1998).
[CrossRef]

F. J. Garcia de Abajo and A. Howie, "Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics," Phys. Rev. Lett. 80, 5180-5183 (1998).
[CrossRef]

1997

N. Zabala, A. Rivacoba, and P. M. Echenique, "Coupling effects in the excitations by an external electron beam near close particles," Phys. Rev. B 56, 7623-7635 (1997).
[CrossRef]

1994

J. B. Pendry and L. Martín-Moreno, "Energy-loss by charged-particles in complex media," Phys. Rev. B 50, 5062-5073 (1994).
[CrossRef]

1992

A. Rivacoba, N. Zabala, and P. M. Echenique, "Theory of energy loss in scanning transmission electron microscopy of supported small particles," Phys. Rev. Lett. 69, 3362-3365 (1992).
[CrossRef] [PubMed]

1985

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, "Excitation of edge modes in the interaction of electron beams with dielectric wedges," Phys. Rev. B 31, 121-126 (1985).
[CrossRef]

T. L. Ferrell, and P. M. Echenique, "Generation of surface excitations on dielectric spheres by an external electron beam," Phys. Rev. Lett. 55, 1526-1529 (1985).
[CrossRef] [PubMed]

1975

P. M. Echenique, and J. B. Pendry, "Absorption profile at surfaces," J. Phys. C 8, 2936-2942 (1975).
[CrossRef]

1957

R. H. Ritchie, "Plasma losses by fast electrons in thin films," Phys. Rev. 106, 874-881 (1957).
[CrossRef]

Alu, A.

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

Cai, W.

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

Chan, C. T.

H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007).
[CrossRef]

Chen, H.

H. Chen and C. T. Chan, "Transformation media that rotate electromagnetic fields," Appl. Phys. Lett. 90, 241105 (2007).
[CrossRef]

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

Chettiar, U. K.

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

Cummer, S. A.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Dong, Y.

J. Xu, Y. Dong, and X. Zhang, "Electromagnetic interactions between a fast electron beam and metamaterial cloaks," Phys. Rev. E 78, 046601 (2008).
[CrossRef]

Echenique, P. M.

F. J. Garcia de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

N. Zabala, A. Rivacoba, and P. M. Echenique, "Coupling effects in the excitations by an external electron beam near close particles," Phys. Rev. B 56, 7623-7635 (1997).
[CrossRef]

A. Rivacoba, N. Zabala, and P. M. Echenique, "Theory of energy loss in scanning transmission electron microscopy of supported small particles," Phys. Rev. Lett. 69, 3362-3365 (1992).
[CrossRef] [PubMed]

T. L. Ferrell, and P. M. Echenique, "Generation of surface excitations on dielectric spheres by an external electron beam," Phys. Rev. Lett. 55, 1526-1529 (1985).
[CrossRef] [PubMed]

P. M. Echenique, and J. B. Pendry, "Absorption profile at surfaces," J. Phys. C 8, 2936-2942 (1975).
[CrossRef]

Engheta, N.

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

Ferrell, T. L.

T. L. Ferrell, and P. M. Echenique, "Generation of surface excitations on dielectric spheres by an external electron beam," Phys. Rev. Lett. 55, 1526-1529 (1985).
[CrossRef] [PubMed]

Garcia de Abajo, F. J.

F. J. Garcia de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

F. J. Garcia de Abajo, "Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam," Phys. Rev. B 59, 3095-3107 (1999).
[CrossRef]

F. J. Garcia de Abajo, "Interaction of radiation and fast electrons with clusters of dielectrics: a multiple scattering approach," Phys. Rev. Lett. 82, 2776-2779 (1999).
[CrossRef]

F. J. Garcia de Abajo and A. Howie, "Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics," Phys. Rev. Lett. 80, 5180-5183 (1998).
[CrossRef]

Garcia-Molina, R.

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, "Excitation of edge modes in the interaction of electron beams with dielectric wedges," Phys. Rev. B 31, 121-126 (1985).
[CrossRef]

Gras-Marti, A.

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, "Excitation of edge modes in the interaction of electron beams with dielectric wedges," Phys. Rev. B 31, 121-126 (1985).
[CrossRef]

Howie, A.

F. J. Garcia de Abajo and A. Howie, "Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics," Phys. Rev. Lett. 80, 5180-5183 (1998).
[CrossRef]

Kildishev, A. V.

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

Kong, J. A.

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Leonhardt, U.

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

Luo, Y.

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Martí, L.

J. B. Pendry and L. Martín-Moreno, "Energy-loss by charged-particles in complex media," Phys. Rev. B 50, 5062-5073 (1994).
[CrossRef]

Miller, D. A. B.

Milton, G. W.

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

Neff, C. W.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

Nellist, P. D.

P. D. Nellist, and S. J. Pennycook, "Subangstrom resolution by underfocused incoherent transmission electron microscopy," Phys. Rev. Lett. 81, 4156-4159 (1998).
[CrossRef]

Nicorovici, N.-A.

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

Ochiai, T.

T. Ochiai, and K. Ohtaka, "Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals. I. Formalism and surface plasmon polariton," Phys. Rev. B 69, 125106 (2004).
[CrossRef]

Ohtaka, K.

T. Ochiai, and K. Ohtaka, "Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals. I. Formalism and surface plasmon polariton," Phys. Rev. B 69, 125106 (2004).
[CrossRef]

Pendry, J. B.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

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]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

J. B. Pendry and L. Martín-Moreno, "Energy-loss by charged-particles in complex media," Phys. Rev. B 50, 5062-5073 (1994).
[CrossRef]

P. M. Echenique, and J. B. Pendry, "Absorption profile at surfaces," J. Phys. C 8, 2936-2942 (1975).
[CrossRef]

Pennycook, S. J.

P. D. Nellist, and S. J. Pennycook, "Subangstrom resolution by underfocused incoherent transmission electron microscopy," Phys. Rev. Lett. 81, 4156-4159 (1998).
[CrossRef]

Popa, B. I.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Qiu, M.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

Ran, L.

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Ritchie, R. H.

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, "Excitation of edge modes in the interaction of electron beams with dielectric wedges," Phys. Rev. B 31, 121-126 (1985).
[CrossRef]

R. H. Ritchie, "Plasma losses by fast electrons in thin films," Phys. Rev. 106, 874-881 (1957).
[CrossRef]

Rivacoba, A.

F. J. Garcia de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

N. Zabala, A. Rivacoba, and P. M. Echenique, "Coupling effects in the excitations by an external electron beam near close particles," Phys. Rev. B 56, 7623-7635 (1997).
[CrossRef]

A. Rivacoba, N. Zabala, and P. M. Echenique, "Theory of energy loss in scanning transmission electron microscopy of supported small particles," Phys. Rev. Lett. 69, 3362-3365 (1992).
[CrossRef] [PubMed]

Ruan, Z.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

Schurig, D.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

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]

Shalaev, V. M.

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

Smith, D. R.

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]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 036621 (2006).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Wu, B.-I.

H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, "Electromagnetic wave interactions with a metamaterial cloak," Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

B. Zhang, H. Chen, B.-I. Wu, Y. Luo, L. Ran, and J. A. Kong, "Response of a cylindrical invisibility cloak to electromagnetic waves," Phys. Rev. B 76, 121101 (2007).
[CrossRef]

Xu, J.

J. Xu, Y. Dong, and X. Zhang, "Electromagnetic interactions between a fast electron beam and metamaterial cloaks," Phys. Rev. E 78, 046601 (2008).
[CrossRef]

J. Xu and X. Zhang, "Relativistic energy loss and induced photon emission in the interaction of a left-handed sphere with an external electron beam," Phys. Lett. A 372, 1129-1134 (2008).
[CrossRef]

Yan, M.

Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, "Ideal cylindrical cloak: perfect but sensitive to tiny perturbations," Phys. Rev. Lett. 99, 113903 (2007).
[CrossRef] [PubMed]

Zabala, N.

F. J. Garcia de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, "Electron energy loss spectroscopy as a probe of two dimensional photonic crystals," Phys. Rev. B 68, 205105 (2003).
[CrossRef]

N. Zabala, A. Rivacoba, and P. M. Echenique, "Coupling effects in the excitations by an external electron beam near close particles," Phys. Rev. B 56, 7623-7635 (1997).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Schematic picture of the geometry depicting a moving electron in vacuum with velocity v in the interaction of a nonideal invisibility cloak cylinder. The outer boundary is still fixed at R 2; the inner boundary is at R 1 + δ, where δ is a positive number.

Fig. 2.
Fig. 2.

Ez field distributions for moving electron beam with v = 0.7c passing outside a dielectric cylinder (b = 1.1R 2) with ε 2/ε 0 = 11.9 + 0.1i and μ 2 = μ 0. (a),(b) Correspond to the case without a cloak in the xy and xz planes, respectively. (c),(d) Correspond to the case with a cloak of δ = 0.5R 1. (e),(f) Correspond to the case with a cloak of δ = 0.01R 1. ε t and μ t of the cloak shell are taken according to Eq. (2), where ε 3 = ε 0 and μ 3 = μ 0. R 1 = 0.5λ 0 and R 2 = 1.0λ 0. Here x, y, and z are in unit of λ 0.

Fig. 3.
Fig. 3.

Energy loss as a function of ω/ω 0 for electron passing outside a dielectric cylinder and a cloak shell with δ = 0.01R 1 and 0.8R 1, respectively. ω 0 = 2πc/λ 0. The other parameters are taken the same as ones in Fig. 2.

Fig. 4.
Fig. 4.

Energy loss as a function of δ for electron passing outside the cloak and cylinder. The other parameters are taken the same as those in Fig. 2.

Fig. 5.
Fig. 5.

Induced Ez field distributions in the xz plane for the electron beam (a) with v = 0.7c moving along the z axis inside the cylindrical air hole without the cloak, (b) with a cloak of δ = 0.5R 1, and (c) δ = 0.01R 1. The ε t and μ t of the cloak shell are taken according to Eq. (2), where ε 3 = ε 0 and μ 3 = μ 0. The dielectric constant of the background or outside the cloak is taken as ε 1/ε 0 = 11.9 + 0.1i and μ 1 = μ 0.

Fig. 6.
Fig. 6.

Energy loss as a function of δ for electrons with v = 0.7c moving along the z axis inside the cylindrical air hole with a nonideal cylindrical cloak. The other parameters are taken identical with those in Fig. 5.

Fig 7.
Fig 7.

Comparison between the loss probability (Γ = Γloss) and the radiation emission probability (Γ = Γrad) as a function of ω/ω 0 for electron beam with v = 0. 7 c moving inside the cloak along the z axis with δ = 0.5R. The ε t and μ t of the cloak are taken according to Eq. (2), where ε 3/ε 0 = 5.0 + 0.2i and μ 3 = μ 0. The background outside the cloak (r > R2)is a dielectric material with ε 1/ε 0 = 11.9 and μ 1 = μ 0. The other parameters are taken the same as those in Fig. 6.

Equations (30)

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ε = ε r r ̂ r ̂ + ε θ θ ̂ θ ̂ + ε z z ̂ z ̂ μ = μ r r ̂ r ̂ + μ θ θ ̂ θ ̂ + μ z z ̂ z ̂ ,
ε r ε 3 = μ r μ 3 = ( r R 1 ) r , ε θ ε 3 = μ θ μ 3 = r ( r R 1 ) ,
ε z ε 3 = μ z μ 3 = R 2 2 ( r R 1 ) r ( R 2 R 1 ) 2
E = E E + E M = c ε 1 × [ μ 1 × ( z ̂ ψ E ( r ) ) ] ε 1 × ( z ̂ ψ M ( r ) ) ,
H = H E + H M = μ 1 × ( z ̂ ψ E ( r ) ) + c μ 1 × [ ε 1 × ( z ̂ ψ M ( r ) ) ] ,
ψ M ( E ) , inc ( r ) = m ψ m M ( E ) , inc J m ( k r 1 r ) e imθ + i k z z ,
ψ M ( E ) , ind ( r ) = ψ m M ( E ) , ind H m ( 1 ) ( k r 1 r ) e imθ + i k z z ,
ψ M ( E ) , int ( r ) = ψ m M ( E ) , int J m ( k r 2 r ) e imθ + i k z z ,
ψ M ( E ) , c ( r ) = m [ ψ m M ( E ) , c 1 J m ( R 2 k r 3 ( r R 1 ) ( R 2 R 1 ) ) + ψ m M ( E ) , c 2 N m ( R 2 k r 3 ( r R 1 ) ( R 2 R 1 ) ) ] e imθ + i k z z ,
ϕ M ( E ) , inc ( r ) = m ϕ m M ( E ) , inc H m ( 1 ) ( k r 2 r ) e imθ + i k z z ,
ϕ M ( E ) , ind ( r ) = ϕ m M ( E ) , ind J m ( k r 2 r ) e imθ + i k z z ,
ϕ M ( E ) , ext ( r ) = ϕ m M ( E ) , ext H m ( 1 ) ( k r 1 r ) e imθ + i k z z ,
ϕ M ( E ) , c ( r ) = m [ ϕ m M ( E ) , c 1 J m ( R 2 k r 3 ( r R 1 ) ( R 2 R 1 ) ) + ϕ m M ( E ) , c 2 N m ( R 2 k r 3 ( r R 1 ) ( R 2 R 1 ) ) ] e imθ + i k z z ,
E inc r ω = [ 1 ε j μ j c 2 v ] πi v m J m ( k rj r < ) H m ( 1 ) ( k rj r > ) e imθ e i k z z ,
ψ m E , inc = μ 1 H m ( 1 ) ( k r 1 b ) c ( b > R 2 ) ,
ϕ m E , inc = μ 2 J m ( k r 2 b ) c ( b < R 1 + δ )
ψ m M , inc = ϕ m M , inc = 0 .
Δ E = d t v E ind r t t = L 0 ω d ω Γ loss ( ω ) ,
Γ loss ( ω ) = 1 πω L d t Re { e iωt v E ind r t ω }
Γ loss ( ω ) = { m Re { k r 1 2 k 1 2 c ψ m E , ind H m ( 1 ) ( k r 1 b ) } ( b > R 2 ) m Re { k r 2 2 k 2 2 c ϕ m E , ind J m ( k r 2 b ) } ( b < R 1 + δ ) ,
Δ E rad = c 4 π L d t d θ d zr [ E r t × H r t ] r ̂ ,
Δ E rad = 0 ω d ω d θ d z Γ rad ( ω , θ , z ) ,
Γ rad ( ω , θ , z ) = cr 4 π 2 ω Re { [ E ( ω ) × H ( ω ) ] r ̂ }
Γ rad ( ω ) = { m ( c k r 1 πω ) 2 ( 1 ε 1 2 μ 1 ψ m M , ind 2 + 1 ε 1 μ 1 2 ψ m E , ind 2 ) , b > R 2 m ( c k r 1 πω ) 2 ( 1 ε 1 2 μ 1 ϕ m M , ext 2 + 1 ε 1 μ 1 2 ϕ m E , ext 2 ) , b < R 1 + δ ,
[ ψ m M , ind ψ m M , c 1 ψ m M , c 2 ψ m M , int ψ m E , ind ψ m E , c 1 ψ m E , c 2 ψ m E , int ] = M 1 [ ψ m E , inc k r 1 2 ε 1 μ 1 J m ( a 1 ) ψ m M , inc k r 1 ε 1 J m ( a 1 ) + ψ m E , inc ζ 1 R 2 J m ( a 1 ) ψ m M , inc k r 1 2 ε 1 μ 1 J m ( a 1 ) ψ m M , inc ζ 1 R 2 J m ( a 1 ) ψ m E , inc k r 1 μ 1 J m ( a 1 ) 0 0 0 0 ] ,
M 11 = [ 0 0 0 0 k r 1 ε 1 H m ( 1 ) ( a 1 ) k r 3 ε 3 J m ( a 3 ) k r 3 ε 3 N m ( a 3 ) 0 k r 1 2 ε 1 μ 1 H m ( 1 ) ( a 1 ) k r 3 2 ε 3 μ 3 J m ( a 3 ) k r 3 2 ε 3 μ 3 N m ( a 3 ) 0 ζ 1 R 2 H m ( 1 ) ( a 1 ) ζ 3 R 2 J m ( a 3 ) ζ 3 R 2 N m ( a 3 ) 0 ] ,
M 12 = [ k r 1 2 ε 1 μ 1 H m ( 1 ) ( a 1 ) k r 3 2 ε 3 μ 3 J m ( a 3 ) k r 3 2 ε 3 μ 3 N m ( a 3 ) 0 ζ 1 R 2 H m ( 1 ) ( a 1 ) ζ 3 R 2 J m ( a 3 ) ζ 3 R 2 N m ( a 3 ) 0 0 0 0 0 k r 1 μ 1 H m ( 1 ) ( a 1 ) k r 3 μ 3 J m ( a 3 ) k r 3 μ 3 N m ( a 3 ) 0 ] ,
M 21 = [ 0 0 0 0 0 a d ε 3 R d J m ( a d ) a d ε 3 R d N m ( a d ) k r 2 ε 2 J m ( a 2 ) 0 k r 3 2 ε 3 μ 3 J m ( a d ) k r 3 2 ε 3 μ 3 N m ( a d ) k r 2 2 ε 2 μ 2 J m ( a 2 ) 0 ζ 3 R d J m ( a d ) ζ 3 R d N m ( a d ) ζ 2 R d J m ( a 2 ) ] ,
M 22 = [ 0 k r 3 2 ε 3 μ 3 J m ( a d ) k r 3 2 ε 3 μ 3 N m ( a d ) k r 2 2 ε 2 μ 2 J m ( a 2 ) 0 ζ 3 R d J m ( a d ) ζ 3 R d N m ( a d ) ζ 2 R d J m ( a 2 ) 0 0 0 0 0 a d μ 3 R d J m ( a d ) a d μ 3 R d N m ( a d ) k r 2 μ 2 J m ( a 2 ) ] .
[ ϕ m M , ext ϕ m M , c 1 ϕ m M , c 2 ϕ m M , ind ϕ m E , ext ϕ m E , c 1 ϕ m E , c 2 ϕ m E , ind ] = M 1 [ ϕ m E , inc k r 2 2 ε 2 μ 2 H m ( 1 ) ( a 2 ) ϕ m M , inc k r 2 ε 2 H m ( 1 ) ( a 2 ) + ϕ m E , inc ζ 2 R d H m ( 1 ) ( a 2 ) ϕ m M , inc k r 2 2 ε 2 μ 2 H m ( 1 ) ( a 2 ) ϕ m M , inc ζ 2 R d H m ( 1 ) ( a 2 ) ϕ m E , inc k r 2 μ 2 H m ( 1 ) ( a 2 ) 0 0 0 0 ] .

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