Abstract

The interactions of two-dimensional 10-fold and 12-fold photonic quasicrystals with a moving electron beam have been studied by using the multiple-scattering method. The electron energy loss spectroscopy (EELS) and three-dimensional local density of states in these systems have been calculated. Some three-dimensional localized states in these defect-free two-dimensional systems have been found. It has been demonstrated that these localized states can be explored by means of the EELS.

© 2009 Optical Society of America

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  7. J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  35. A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021-10027 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]

2009

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]

F. J. Garcıia de Abajo, and M. Kociak, "Probing the Photonic Local Density of States with Electron Energy Loss Spectroscopy," Phys. Rev. Lett. 100, 106804 (2008).
[CrossRef]

2007

K. Mnaymneh and R. C. Gauthier, Mode localization and band-gap formation in defect-free photonic quasicrystalsOpt. Express 14, 5089-5099 (2007).
[CrossRef]

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

2006

2005

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Z. Feng, X. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, and D.Z. Zhang, "Negative refraction and imaging using 12-fold-symmetry quasicrystals," Phys. Rev. Lett. 94, 247402 (2005).
[CrossRef]

2004

D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, "Three-dimensional Green’s tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders," Phys. Rev. E 70, 066608 (2004).
[CrossRef]

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.G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, "Cherenkov effect as a probe of photonic nanostructures," Phys. Rev. Lett. 91, 143902 (2003).
[CrossRef] [PubMed]

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]

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

2001

X. Zhang, Z.-Q. Zhang, and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, R081105 (2001).
[CrossRef]

2000

M. E. Zoorob, M. D. B. Charleton, G. J. Parker, J. J. Baumberg, and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740-743 (2000).
[CrossRef] [PubMed]

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. Garcıia 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

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]

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

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic Band Gap Guidance in Optical Fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956-959 (1998).
[CrossRef]

1997

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

1994

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band Gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

J. B. Pendry and L. Martın-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 miceoscopy of supported small particles," Phys. Rev. Lett. 69, 3362-3365 (1992).
[CrossRef] [PubMed]

1991

S. John and J. Wang, "Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms," Phys. Rev. Lett. 64, 2418-2421 (1991).
[CrossRef]

1989

N. Zabala, A. Rivacoba, and P. M. Echenique, "Energy loss of electrons traveling through cylindrical holes," Surf. Sci. 209, 465-480 (1989).
[CrossRef]

1988

B. L. Illman, V. E. Anderson, R. J. Warmack, and T. L. Ferrell, "Spectrum of surface-mode contributions to the differential energy-loss probability for electrons passing by a spheroid," Phys. Rev. B 38, 3045-3049 (1988).
[CrossRef]

1987

T. L. Ferrell, R. J. Warmack, V. E. Anderson, and P. M. Echenique, "Analytical calculation of stopping power for isolated small spheres," Phys. Rev. B 35, 7365-7371 (1987).
[CrossRef]

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[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]

Anderson, V. E.

B. L. Illman, V. E. Anderson, R. J. Warmack, and T. L. Ferrell, "Spectrum of surface-mode contributions to the differential energy-loss probability for electrons passing by a spheroid," Phys. Rev. B 38, 3045-3049 (1988).
[CrossRef]

T. L. Ferrell, R. J. Warmack, V. E. Anderson, and P. M. Echenique, "Analytical calculation of stopping power for isolated small spheres," Phys. Rev. B 35, 7365-7371 (1987).
[CrossRef]

Baumberg, J. J.

M. E. Zoorob, M. D. B. Charleton, G. J. Parker, J. J. Baumberg, and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740-743 (2000).
[CrossRef] [PubMed]

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic Band Gap Guidance in Optical Fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Bloemer, M. J.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band Gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Bowden, C. M.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band Gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic Band Gap Guidance in Optical Fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Capolino, F.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021-10027 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Chan, C. T.

X. Zhang, Z.-Q. Zhang, and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, R081105 (2001).
[CrossRef]

Chan, C.T.

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956-959 (1998).
[CrossRef]

Chan, Y. S.

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956-959 (1998).
[CrossRef]

Charleton, M. D. B.

M. E. Zoorob, M. D. B. Charleton, G. J. Parker, J. J. Baumberg, and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740-743 (2000).
[CrossRef] [PubMed]

Cheng, B.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Cheng, B.Y.

Z. Feng, X. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, and D.Z. Zhang, "Negative refraction and imaging using 12-fold-symmetry quasicrystals," Phys. Rev. Lett. 94, 247402 (2005).
[CrossRef]

Colliex, C.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Della-Villa, A.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021-10027 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

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]

Dowling, J. P.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band Gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Echenique, P. M.

F. J. Garcia de Abajo, A.G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, "Cherenkov effect as a probe of photonic nanostructures," Phys. Rev. Lett. 91, 143902 (2003).
[CrossRef] [PubMed]

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]

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

N. Zabala, A. Rivacoba, and P. M. Echenique, "Energy loss of electrons traveling through cylindrical holes," Surf. Sci. 209, 465-480 (1989).
[CrossRef]

T. L. Ferrell, R. J. Warmack, V. E. Anderson, and P. M. Echenique, "Analytical calculation of stopping power for isolated small spheres," Phys. Rev. B 35, 7365-7371 (1987).
[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]

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

Enoch, S.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021-10027 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Feng, Z.

Z. Feng, X. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, and D.Z. Zhang, "Negative refraction and imaging using 12-fold-symmetry quasicrystals," Phys. Rev. Lett. 94, 247402 (2005).
[CrossRef]

Ferrell, T. L.

B. L. Illman, V. E. Anderson, R. J. Warmack, and T. L. Ferrell, "Spectrum of surface-mode contributions to the differential energy-loss probability for electrons passing by a spheroid," Phys. Rev. B 38, 3045-3049 (1988).
[CrossRef]

T. L. Ferrell, R. J. Warmack, V. E. Anderson, and P. M. Echenique, "Analytical calculation of stopping power for isolated small spheres," Phys. Rev. B 35, 7365-7371 (1987).
[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]

Fussell, D. P.

D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, "Three-dimensional Green’s tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders," Phys. Rev. E 70, 066608 (2004).
[CrossRef]

Galdi, V.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021-10027 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Garcia de Abajo, F.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

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, A.G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, "Cherenkov effect as a probe of photonic nanostructures," Phys. Rev. Lett. 91, 143902 (2003).
[CrossRef] [PubMed]

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]

Garciia de Abajo, F. J.

F. J. Garcıia de Abajo, and M. Kociak, "Probing the Photonic Local Density of States with Electron Energy Loss Spectroscopy," Phys. Rev. Lett. 100, 106804 (2008).
[CrossRef]

F. J. Garcıia 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]

Gauthier, R. C.

K. Mnaymneh and R. C. Gauthier, Mode localization and band-gap formation in defect-free photonic quasicrystalsOpt. Express 14, 5089-5099 (2007).
[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]

Henrard, L.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[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]

Hu, X.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Illman, B. L.

B. L. Illman, V. E. Anderson, R. J. Warmack, and T. L. Ferrell, "Spectrum of surface-mode contributions to the differential energy-loss probability for electrons passing by a spheroid," Phys. Rev. B 38, 3045-3049 (1988).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

John, S.

S. John and J. Wang, "Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms," Phys. Rev. Lett. 64, 2418-2421 (1991).
[CrossRef]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Knight, J. C.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic Band Gap Guidance in Optical Fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Kociak, M.

F. J. Garcıia de Abajo, and M. Kociak, "Probing the Photonic Local Density of States with Electron Energy Loss Spectroscopy," Phys. Rev. Lett. 100, 106804 (2008).
[CrossRef]

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Li, Z.Y.

Z. Feng, X. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, and D.Z. Zhang, "Negative refraction and imaging using 12-fold-symmetry quasicrystals," Phys. Rev. Lett. 94, 247402 (2005).
[CrossRef]

Liu, Z. Y.

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956-959 (1998).
[CrossRef]

Liz-Marzan, L. M.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Martijn de Sterke, C.

D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, "Three-dimensional Green’s tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders," Phys. Rev. E 70, 066608 (2004).
[CrossRef]

Martin-Moreno, L.

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

McPhedran, R. C.

D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, "Three-dimensional Green’s tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders," Phys. Rev. E 70, 066608 (2004).
[CrossRef]

Mnaymneh, K.

K. Mnaymneh and R. C. Gauthier, Mode localization and band-gap formation in defect-free photonic quasicrystalsOpt. Express 14, 5089-5099 (2007).
[CrossRef]

Nelayah, J.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

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]

Netti, M. C.

M. E. Zoorob, M. D. B. Charleton, G. J. Parker, J. J. Baumberg, and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740-743 (2000).
[CrossRef] [PubMed]

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]

Parker, G. J.

M. E. Zoorob, M. D. B. Charleton, G. J. Parker, J. J. Baumberg, and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740-743 (2000).
[CrossRef] [PubMed]

Pastoriza-Santos, I.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Pattantyus-Abraham, A.G.

F. J. Garcia de Abajo, A.G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, "Cherenkov effect as a probe of photonic nanostructures," Phys. Rev. Lett. 91, 143902 (2003).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry and L. Martın-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]

Pierro, V.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021-10027 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

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.G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, "Cherenkov effect as a probe of photonic nanostructures," Phys. Rev. Lett. 91, 143902 (2003).
[CrossRef] [PubMed]

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]

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

N. Zabala, A. Rivacoba, and P. M. Echenique, "Energy loss of electrons traveling through cylindrical holes," Surf. Sci. 209, 465-480 (1989).
[CrossRef]

Russell, P. St. J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic Band Gap Guidance in Optical Fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Scalora, M.

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band Gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Stephan, O.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Taverna, D.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Tayeb, G.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021-10027 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Tence, M.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Wang, J.

S. John and J. Wang, "Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms," Phys. Rev. Lett. 64, 2418-2421 (1991).
[CrossRef]

Wang, Y.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Wang, Y.Q.

Z. Feng, X. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, and D.Z. Zhang, "Negative refraction and imaging using 12-fold-symmetry quasicrystals," Phys. Rev. Lett. 94, 247402 (2005).
[CrossRef]

Warmack, R. J.

B. L. Illman, V. E. Anderson, R. J. Warmack, and T. L. Ferrell, "Spectrum of surface-mode contributions to the differential energy-loss probability for electrons passing by a spheroid," Phys. Rev. B 38, 3045-3049 (1988).
[CrossRef]

T. L. Ferrell, R. J. Warmack, V. E. Anderson, and P. M. Echenique, "Analytical calculation of stopping power for isolated small spheres," Phys. Rev. B 35, 7365-7371 (1987).
[CrossRef]

Wolf, M. O.

F. J. Garcia de Abajo, A.G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, "Cherenkov effect as a probe of photonic nanostructures," Phys. Rev. Lett. 91, 143902 (2003).
[CrossRef] [PubMed]

Xu, J.

J. Xu and X. Zhang, "Cloaking radiation of moving electron beam and relativistic energy loss spectra," Opt. Express 17, 4758-4772 (2009).
[CrossRef] [PubMed]

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

Xu, X.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Zabala, N.

F. J. Garcia de Abajo, A.G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, "Cherenkov effect as a probe of photonic nanostructures," Phys. Rev. Lett. 91, 143902 (2003).
[CrossRef] [PubMed]

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]

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

N. Zabala, A. Rivacoba, and P. M. Echenique, "Energy loss of electrons traveling through cylindrical holes," Surf. Sci. 209, 465-480 (1989).
[CrossRef]

Zhang, D.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Zhang, D.Z.

Z. Feng, X. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, and D.Z. Zhang, "Negative refraction and imaging using 12-fold-symmetry quasicrystals," Phys. Rev. Lett. 94, 247402 (2005).
[CrossRef]

Zhang, X.

J. Xu and X. Zhang, "Cloaking radiation of moving electron beam and relativistic energy loss spectra," Opt. Express 17, 4758-4772 (2009).
[CrossRef] [PubMed]

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

Z. Feng, X. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, and D.Z. Zhang, "Negative refraction and imaging using 12-fold-symmetry quasicrystals," Phys. Rev. Lett. 94, 247402 (2005).
[CrossRef]

X. Zhang, Z.-Q. Zhang, and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, R081105 (2001).
[CrossRef]

Zhang, Z.-Q.

X. Zhang, Z.-Q. Zhang, and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, R081105 (2001).
[CrossRef]

Zoorob, M. E.

M. E. Zoorob, M. D. B. Charleton, G. J. Parker, J. J. Baumberg, and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740-743 (2000).
[CrossRef] [PubMed]

J. Phys. C

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

Nat. Phys.

J. Nelayah, M. Kociak, O. Stephan, F. Garcia de Abajo, M. Tence, L. Henrard, D. Taverna, I. Pastoriza-Santos, L. M. Liz-Marzan, and C. Colliex, "Mapping surface plasmons on a single metallic nanoparticle," Nat. Phys. 3, 348-353 (2007).
[CrossRef]

Nature

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

M. E. Zoorob, M. D. B. Charleton, G. J. Parker, J. J. Baumberg, and M. C. Netti, "Complete photonic bandgaps in 12-fold symmetric quasicrystals," Nature 404, 740-743 (2000).
[CrossRef] [PubMed]

Opt. Express

Phys. Rev.

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

Phys. Rev. B

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]

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

X. Zhang, Z.-Q. Zhang, and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, R081105 (2001).
[CrossRef]

B. L. Illman, V. E. Anderson, R. J. Warmack, and T. L. Ferrell, "Spectrum of surface-mode contributions to the differential energy-loss probability for electrons passing by a spheroid," Phys. Rev. B 38, 3045-3049 (1988).
[CrossRef]

T. L. Ferrell, R. J. Warmack, V. E. Anderson, and P. M. Echenique, "Analytical calculation of stopping power for isolated small spheres," Phys. Rev. B 35, 7365-7371 (1987).
[CrossRef]

F. J. Garcıia 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]

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

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]

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]

Phys. Rev. E

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

D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, "Three-dimensional Green’s tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders," Phys. Rev. E 70, 066608 (2004).
[CrossRef]

Phys. Rev. Lett.

F. J. Garcıia de Abajo, and M. Kociak, "Probing the Photonic Local Density of States with Electron Energy Loss Spectroscopy," Phys. Rev. Lett. 100, 106804 (2008).
[CrossRef]

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

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

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956-959 (1998).
[CrossRef]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Z. Feng, X. Zhang, Y.Q. Wang, Z.Y. Li, B.Y. Cheng, and D.Z. Zhang, "Negative refraction and imaging using 12-fold-symmetry quasicrystals," Phys. Rev. Lett. 94, 247402 (2005).
[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]

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]

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, A.G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, "Cherenkov effect as a probe of photonic nanostructures," Phys. Rev. Lett. 91, 143902 (2003).
[CrossRef] [PubMed]

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

S. John and J. Wang, "Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms," Phys. Rev. Lett. 64, 2418-2421 (1991).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, "Optical limiting and switching of ultrashort pulses in nonlinear photonic band Gap materials," Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Science

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic Band Gap Guidance in Optical Fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Surf. Sci.

N. Zabala, A. Rivacoba, and P. M. Echenique, "Energy loss of electrons traveling through cylindrical holes," Surf. Sci. 209, 465-480 (1989).
[CrossRef]

Other

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal-Molding the Flow of Light, (Princeton University Press, Princeton, NJ, 1995).

C. M. Soukoulis, Photonic Band Gap Materials, (Kluwer, Academic, Dordrecht,1996).

K. Sakoda, Optical properties of photonic crystals, (Springer, 2001).

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Figures (5)

Fig. 1.
Fig. 1.

Energy loss spectra for an electron moving with v=0.7c along the axis of a cylindrical hole (b=0.0) in the background with ε=11.4+0.1i. (a) 10-fold QCs with diameter D=0.58a for various size of the sample. Here N represents the number of the rods in the sample; (b) 12-fold QCs with diameter D=0.58a for various size of the sample; (c) The periodic structure with a triangular lattice and diameter D=0.8a. Inset shows the structure of the system. The spectra is given in units of 2×103 cP(ω). And consecutive curves are shifted 1 unit upward for QCs and 0.5 unit for periodic structure for readability.

Fig. 2.
Fig. 2.

Energy loss spectra for 12-fold QCs with N=55 as a function of the frequency with different impact parameters (a) and different electron velocities (b). The other parameters are identical with those in Fig.1(a).

Fig. 3.
Fig. 3.

(a). Comparison between EELS P(ω) and LDOS ρ ˜ z for 12-fold QCs with different size as a function of the frequency. The z-projected LDOS is given in units of (π/2ω) ρ ˜ z . (b) The corresponding result with different electron velocities at N=55. (c) The distribution of eigen-electric field (|Ez |) inside the sample with N=55 at ωa/2πc=0.318. The field (Ez ) map is in linear false-color scale (red=high; blue=low). The other parameters are identical with those in Fig. 1 (b).

Fig. 4.
Fig. 4.

(a). Comparison between EELS P(ω) and LDOS ρ ˜ z for 10-fold QCs with different size as a function of the frequency at v=0.7c. (b). The corresponding result with different electron velocities at N=141. (c). The distribution of eigen-electric field (|Ez |) inside the sample with N=141 at ωa/2πc=0.324. The field (|Ez |) map is in linear false-color scale (red=high; blue=low). The other parameters are identical with those in Fig. 1(a).

Fig. 5.
Fig. 5.

Relation between EELS and LDOS for 10-fold QCs with N=51 as function of the impact parameter b and the frequency. Those maps are in linear false-color scale (red=high; blue=low). (a) EELS P(ω) ; (b) z-projected LDOS (−π/2ω) ρ ˜ z ; (c) unprojected LDOS (−π/2ω) ρ ˜ . The other parameters are the same to those in Fig. 3.

Equations (40)

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Δ E = d t v · E ind ( r t , t ) = L 0 ω d ω P ( ω ) ,
P ( ω ) = 1 π ω L dt Re { e i ω t v · E ind ( r t , ω ) } .
E ind ( r , ω ) = Σ m ( ψ l , ms in E l , β ms J ( r ) + ψ l , mp in E l , β mp J ( r ) ) ,
E i , β ms J ( r ) = [ im k R R J m ( k R R ) R ̂ J m ( k R R ) φ ̂ ] e im φ e i β z ,
E l , β mp J ( r ) = β kn ( r ) [ i j m ( k R R ) R ̂ m k R R J m ( k R R ) φ ̂ + k R β J m ( k R R ) z ̂ ] e im φ e i β z ,
[ ψ i , ms in ψ i , mp in ] = [ m i , ss m i , sp m i , ps m i , pp ] { [ m i , ss m i , sp m i , ps m i , pp ] 1 { [ ψ i , ms ψ i , mp ] [ m i , ss m i , sp m i , ps m i , pp ] [ 0 Q m ext ] } } .
+ [ m i , ss m i , sp m i , ps m i , pp ] [ 0 Q m ext ]
ψ i , ms = δ il m l , sp Q m ext + Σ j = 1 j i n Σ m = Π mm ij ( m j , ss ψ j , ms + m j , sp ψ j , mp )
ψ i , mp = δ il m l , pp Q m ext + Σ j = 1 j i n Σ m = Π mm ij ( m j , ps ψ j , ms + m j , pp ψ j , mp )
Π mm ij = H m m ( 1 ) ( k R R ij ) e i ( m m ) φ ij .
ρ ( r , ω ) = 2 ω π Im { Tr [ G E ( r , r , ω ) ] } .
G E ( r , r , ω ) = 1 2 π d β e i β ( z z ) G ˜ E ( R , R , β , ω ) ,
G ˜ E ( R , R , β , ω ) = ( G ˜ xx E G ˜ xy E G ˜ xz E G ˜ yx E G ˜ yy E G ˜ yz E G ˜ zx E G ˜ zy E G ˜ zz E ) .
( R 2 + k R 2 ) G ˜ zu V ( R , R , β , ω ) = D u v δ ( R , R )
D u E = δ zu + i β k 2 n ( r ) 2 · u ̂ and D u H = z ̂ · × u ̂ .
G ˜ zu l , V ( R , R , β , ω ) = i 4 D u V { H 0 ( 1 ) ( k R R R ) } + Σ m = C zu , m l , V J m ( k R R ) e im ϕ .
B zu , m i , E = δ il ( T m l , ss Q zu , m E + T m l , sp Q zu , m H ) + Σ j = 1 j i n Σ m = S mm ij ( R m j , ss B zu , m j , E + R m j , sp B zu , m j , H )
B zu , m i , H = δ il ( T m l , ps Q zu , m E + T m l , pp Q zu , m H ) + Σ j = 1 j i n Σ m = S mm ij ( R m j , ps B zu , m j , E + R m j , pp B zu , m j , H )
S mm ij = H m m ( 1 ) ( k R R ij ) e i ( m m ) φ ij .
ρ ˜ z ( R , β , ω ) = 2 ω π Im [ z ̂ · G ˜ E ( R , R , β , ω ) · z ̂ ] = 2 ω π Im [ G ˜ zz E ( R , R , β , ω ) ] .
m ss = 1 δ m [ ( k 2 α H + J k + 2 α J H + ) ( α J J + α J + J ) m 2 τ 2 J + H + J 2 ] ,
m sp = 1 δ m m τ J 2 2 ik + k R π k R + 2 a 0 = + ps ,
m pp = 1 δ m [ ( k 2 α J + J k + 2 α J J + ) ( α J H + α H + J ) m 2 τ 2 J + H + J 2 ] ,
m ps = 1 δ m m τ J 2 2 i k + k R π k R + 2 a 0 .
m ps = 1 δ m m τ 2 i k + J H + π k R a 0 ,
m pp = 1 δ m ( α J H + α H + J ) 2 i k k + π k R + a 0 ,
m sp = 1 δ m m τ 2 i k J H + π k R a 0 ,
m ss = 1 δ m ( k 2 α H + J k + 2 α J H + ) 2 i π k R a 0 .
m ss = 1 δ m [ ( k 2 α H + J k + 2 α J H + ) ( α H H + α H + H ) m 2 τ 2 J H H + 2 ] ,
m sp = 1 δ m m τ H + 2 2 i k k R + π k R 2 a 0 ,
m ps = 1 δ m m τ H + 2 2 i k k R + π k R 2 a 0 ,
m pp = 1 δ m [ ( k 2 α H + J k + 2 α J H + ) ( α J H + α H + J ) m 2 τ 2 J H H + 2 ] .
m ps = 1 δ m m τ 2 i k J H + π k R + a 0 ,
m pp = 1 δ m ( α J H + α H + J ) 2 i k k + π k R + a 0 ,
m sp = 1 δ m m τ 2 i k + J H + π k R + a 0 ,
m ss = 1 δ m ( k 2 α H + J k + 2 α J H + ) 2 i π k + a 0 .
τ = ( k R 2 k R + 2 ) β k R + k R a 0 ,
δ m = m 2 τ 2 J 2 H + 2 + ( α H + J α J H + ) ( k 2 α J H + k + 2 α H + J ) ,
α J H + = J m H m + k R + ,
J m ± = J m ( k R ± a 0 ) .

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