## Interaction of fast electron beam with photonic quasicrystals

Optics Express, Vol. 17, Issue 16, pp. 13270-13282 (2009)

http://dx.doi.org/10.1364/OE.17.013270

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### 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

## 1. Introduction

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

2. R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. **106**, 874–881 (1957). [CrossRef]

17. 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]

2. R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. **106**, 874–881 (1957). [CrossRef]

3. P. M. Echenique and J. B. Pendry, “Absorption profile at surfaces,” J. Phys. C **8**, 2936–2942 (1975). [CrossRef]

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

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

6. 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]

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). [CrossRef]

8. 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]

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

12. 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]

13. 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]

13. 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]

19. 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]

29. 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]

33. 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]

34. 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]

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

## 2. Electron energy loss spectroscopy in 2D photonic QCs

*L*is the length of the trajectory, and the electron trajectory is

*r*⃑

*=(*

_{t}*b*,0,

*vt*),

*b*is the impact parameters,

*v*is the velocity of the electron, and

*ω*. Here

*E*⃑

*represents the induced scattered field. Notice that only the z component of the induced field parallel to the velocity of the electron is needed. In this paper, we consider the cavities in mediums, so the scattered field inside the*

^{ind}*l*th cylinder can be expressed as

*s*polarization, and

*p*polarization by using cylindrical coordinates

*r*⃗=(

*R*,

*φ*,

*z*), m is the azimuthal quantum number, and

*z*axis,

*k*is the wave number of the light in the vacuum,

*n*(

*r*⃗) is the refractive index of the martial. The Eqs. (4) and (5) are defined as standing waves, the outgoing waves

*E*,

^{H}_{i}_{βmσ}(

*r*⃑) (σ(σ′)=

*s*,

*p*) can be obtained by substituting the Hankel function

*H*

^{(1)}

*for the Bessel function*

_{m}*J*in the expressions. With this choice, the outgoing waves vanish in the far-field limit. The coefficients {ψ

_{m}*,ψ*

^{in}_{l,ms}*} represent the field inside the cylinders, and they can be obtained by using the boundary condition which requires the tangential components of the field to be continuous at the surface of the cylinder. The process yields the following relation [18*

^{in}_{l,mp}18. 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]

*ℜ*

^{i,σσ′}

*and ℑ*

_{m}^{i,σσ′}

*are the matrix elements of the external reflection and transmission matrices of the*

_{m}*i*th cylinder, ℜ′

^{i,σσ′}

*and ℑ′*

_{m}^{i,σσ′}

*are the matrix elements of the internal reflection and transmission matrices. The explicit expressions for them are given in Appendix. The*

_{m}18. 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]

*ε*is the relative permittivity. Here {ψ

*,ψ*

^{i,ms}*} represents the scattered coefficient outside the cylinders, which can be obtained by solving the following self-consistent equation:*

^{i,mp}*δ*is taken as 1 for

_{il}*i*=

*l*and 0 for the otherwise, (

*R*,

_{ij}*φ*) are the local polar coordinates of the vector

_{ij}*R*⃑

*≡*

_{ij}*R*⃑

*−*

_{j}*R*⃑

*,*

_{i}*R*⃑

*is the position of the*

_{i}*i*th cylinder. Based on Eqs. (1)–(9), the EELS for the 2D QCs can be obtained by the numerical calculations.

18. 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]

*D*=0.58

*a*, a is the lattice constant. The velocity of the moving electron beam is

*v*=0.7

*c*which corresponds to 200-keV, where c is the velocity of the light in the vacuum. The sizes of the sample under consideration are shown as labels attached to the different curves. The N represents the number of the air rods in the sample. Here we consider the electrons move along the middle air rod of the sample (

*b*=0.0). For comparison, we also plot the corresponding results for the periodic structure with a triangular lattice and diameter

*D*=0.8

*a*in Fig. 1(c). For the periodic structure, the band-gap feature can exhibit in the EELS, which is in agreement with those in Ref. [18

**68**, 205105 (2003). [CrossRef]

*v*=0.7

*c*. The number of the rods is N=55. It is shown clearly that both of the position and the number of the peaks do not change with the change of the impact parameter. In contrast, they depend on the velocity of the moving electron. The variations of the loss probabilities with electron velocity for the corresponding case are given in Fig. 2(b). Here the velocity of the electron is taken as 0.55

*c*, 0.7

*c*and 0.78

*c*, corresponding to 100keV, 200keV and 300keV, respectively. As the velocity of the moving electron increases, the scaled loss probability becomes larger, and the changes of the position and number of the peak have also been observed. This is because the velocity change of the electron beam results in the changes of the momentum along the z axis (β=ω/

*v*) and the in-plane component of the wave vector

17. 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]

**68**, 205105 (2003). [CrossRef]

15. 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]

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

*Al*

_{2}

*O*

_{3}, similar phenomena can also be found.

## 3. Local density of states in 2D photonic QCs

37. 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]

37. 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]

*G*(

^{E}*r*⃑,

*r*⃑′,ω) is the electromagnetic Green function with a source at the location

*r*⃗′ and observation point at

*r*⃗, which can be obtained by applying the following inverse Fourier transformation:

*G*̃

*represents the*

^{E}_{uo}*u*(

*u*(

*o*)=

*x*,

*y*,

*z*) component of the electric field vector generated by a source radiating parallel to

*o*axis. The z-components of the fields for a source oriented in the direction of the unit vector

*û*are determined by

*V*={

*E*,

*H*}. The transverse components of the field are straightforwardly obtained from the z-components by using the Maxwell’s equation. Thus, the field expansion inside

*l*th cylinder can be expressed as

*C*,

^{l,E}_{zu,m}*C*}, can be obtained by the scattered coefficients of the system {

^{l,H}_{zu,m}*B*,

^{i,E}_{zu,m}*B*} [37

^{i,H}_{zu,m}37. 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]

*Q*,

^{E}_{zu,m}*Q*} are the results of the point source. And

^{H}_{zu,m}*R*

^{j,σσ′}

*and*

_{m}*T*

^{l,σσ′}

*are the matrix elements of the reflection and transmission matrices [37*

_{m}**70**, 066608 (2004). [CrossRef]

*z*component of the LDOS is

34. 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]

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

*E*|) in the sample are calculated. The calculated result at ω

_{z}*a*/2π

*c*=0.318 is plotted in Fig. 3(c). The localized feature of the eigen-field is shown clearly. The phenomenon is not only observed in 12-fold

*E*|) in Fig. 4(c) corresponds to the resonant peak in Fig. 4(a) at ω

_{z}*a*/2π

*c*=0.324. This means that the peaks marked by the arrows in Figs. 3(a) and 4(a) actually represent the defect-free localized states. However, these states are only localized in 2D plane vertical to the axis of the cylinder, which depend on the momentum along the cylinder axis. It is interesting that such localized states can display in the EELS.

*b*=0.0. In fact, if we change the impact parameter, the corresponding can be still observed. In Fig. 5, we offer a more detailed comparison between the EELS and LDOS with different impact parameters

*b*for the 10-fold QCs with N=51. Figure 5(a) corresponds to the EELS as a function the frequency and the impact parameter, 5(b) to the z-projected LDOS and 5(c) to the unprojected LDOS (

*+*

_{x}*+*

_{y}*). It is seen clearly that the corresponding among them is very well. This means that the localized states in the QCs can be explored by using EELS with different impact parameters for β>ω/*

_{z}*c*.

## 4. Conclusion

## Appendix

*k*

^{+}and

*k*

^{−}are the wave vectors outside and inside the cylinder, respectively,

## Acknowledgments

## References and links

1. | P. D. Nellist and S. J. Pennycook, “Subangstrom resolution by underfocused incoherent transmission electron microscopy,” Phys. Rev. Lett. |

2. | R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. |

3. | P. M. Echenique and J. B. Pendry, “Absorption profile at surfaces,” J. Phys. C |

4. | N. Zabala, A. Rivacoba, and P. M. Echenique, “Energy loss of electrons traveling through cylindrical holes,” Surf. Sci. |

5. | J. Xu and X. Zhang, “Cloaking radiation of moving electron beam and relativistic energy loss spectra,” Opt. Express |

6. | 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 |

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. |

8. | T. L. Ferrell and P. M. Echenique, “Generation of surface excitations on dielectric spheres by an external electron beam,” Phys. Rev. Lett. |

9. | 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 |

10. | F. J. Garciia de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B |

11. | J. Xu, Y. Dong, and X. Zhang, “Electromagnetic interactions between a fast electron beam and metamaterial cloaks,” Phys. Rev. E |

12. | 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 |

13. | A. Rivacoba, N. Zabala, and P. M. Echenique, “Theory of energy loss in scanning transmission electron miceoscopy of supported small particles,” Phys. Rev. Lett. |

14. | J. B. Pendry and L. Martin-Moreno, “Energy-loss by charged-particles in complex media,” Phys. Rev. B |

15. | F. J. Garcia de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. |

16. | F. J. Garcia de Abajo, “Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach,” Phys. Rev. Lett. |

17. | 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. |

18. | 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 |

19. | 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 |

20. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn |

21. | C. M. Soukoulis, |

22. | K. Sakoda, |

23. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

24. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

25. | S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. |

26. | 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. |

27. | J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic Band Gap Guidance in Optical Fibers,” Science |

28. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature |

29. | Y. S. Chan, C.T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. |

30. | 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 |

31. | X. Zhang, Z.-Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B |

32. | 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. |

33. | 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. |

34. | Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, “Localized modes in defect-free dodecagonal quasiperiodic photonic crystals,” Phys. Rev. B |

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 |

36. | K. Mnaymneh and R. C. Gauthier, Mode localization and band-gap formation in defect-free photonic quasicrystalsOpt. Express |

37. | 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 |

38. | F. J. Garciia de Abajo and M. Kociak, “Probing the Photonic Local Density of States with Electron Energy Loss Spectroscopy,” Phys. Rev. Lett. |

**OCIS Codes**

(160.4670) Materials : Optical materials

(260.2110) Physical optics : Electromagnetic optics

(300.2140) Spectroscopy : Emission

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 10, 2009

Revised Manuscript: July 17, 2009

Manuscript Accepted: July 17, 2009

Published: July 20, 2009

**Citation**

Wei Zhong, Jinying Xu, and Xiangdong Zhang, "Interaction of fast electron beam with photonic quasicrystals," Opt. Express **17**, 13270-13282 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13270

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### References

- P. D. Nellist and S. J. Pennycook, "Subangstrom resolution by underfocused incoherent transmission electron microscopy," Phys. Rev. Lett. 81, 4156-4159 (1998). [CrossRef]
- R. H. Ritchie, "Plasma losses by fast electrons in thin films," Phys. Rev. 106, 874-881 (1957). [CrossRef]
- P. M. Echenique and J. B. Pendry, "Absorption profile at surfaces," J. Phys. C 8, 2936-2942 (1975). [CrossRef]
- N. Zabala, A. Rivacoba, and P. M. Echenique, "Energy loss of electrons traveling through cylindrical holes," Surf. Sci. 209, 465-480 (1989). [CrossRef]
- J. Xu and X. Zhang, "Cloaking radiation of moving electron beam and relativistic energy loss spectra," Opt. Express 17, 4758-4772 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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. Xu, Y. Dong, and X. Zhang, "Electromagnetic interactions between a fast electron beam and metamaterial cloaks," Phys. Rev. E 78, 046601 (2008). [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]
- 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]
- 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 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]
- 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]
- 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).
- 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]
- 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]
- J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997). [CrossRef]
- 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]
- 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]
- 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]
- 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]
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