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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13270–13282
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Interaction of fast electron beam with photonic quasicrystals

Wei Zhong, Jinying Xu, and Xiangdong Zhang  »View Author Affiliations


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

Scanning transmission electron microscopy (STEM) has proved to be a powerful technique for determining different microstructures of nanometer scale [1

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

]. The interaction between the moving electron and the microstructures gives rise to the emission and excitation of the radiation. Thus, the electron energy loss spectroscopy (EELS) in the STEM is also a useful tool to investigate both surface and bulk excitations of the samples [2

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

17

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]

]. In the last decades a remarkable progress has been made on this subject. At the beginning, some analytical descriptions of EELS have been limited to simple geometries such as planes [2

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

, 3

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

], cylinders [4

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

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]

], hyperbolic wedges [6

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]

], triangles [7

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]

], spheres[8

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

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]

], spheroid [12

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]

] or combination of spherical interfaces and a plane[13

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]

]. Recently, the discussions have been extended to more complex geometries such as photonic crystal (PCs) [13

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

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]

].

In fact, the gap exists not only in the periodic structures, but also in some photonic quasi-periodic structures. Recently, the transport properties of the wave at various photonic quasicrystals (QCs) have been investigated [29

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

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]

]. The results show that the photonic QCs posses some special properties which there does not exist in the periodic PCs, for example, the localized states can occur in defect-free photonic QCs [34

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

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

]. Then, the problem is whether or not these unusual properties in various quasi-periodic structures can be explored by the STEM?

Considering such a problem, in this paper we investigate the interaction between a fast electron beam and 2D photonic QCs based on exact multiple-scattering method. The energy-loss spectra caused by a fast electron beam moving inside the 2D photonic QCs with the different combinations of electron velocity and impact parameter will be calculated. The local density of states (LDOS) for the systems will be analyzed. The relationships among the EELS, LDOS and localized states will be also discussed.

2. Electron energy loss spectroscopy in 2D photonic QCs

ΔE=dtv·Eind(rt,t)=L0ωdωP(ω),
(1)

where L is the length of the trajectory, and the electron trajectory is rt=(b,0,vt),b is the impact parameters, v is the velocity of the electron, and

P(ω)=1πωLdtRe{eiωtv·Eind(rt,ω)}.
(2)

represents the electron energy loss probability (EELS) per unit of path length in frequency space ω. Here Eind 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 l th cylinder can be expressed as

Eind(r,ω)=Σm(ψl,msinEl,βmsJ(r)+ψl,mpinEl,βmpJ(r)),
(3)

where

Ei,βmsJ(r)=[imkRRJm(kRR)R̂Jm(kRR)φ̂]eimφeiβz,
(4)

for s polarization, and

El,βmpJ(r)=βkn(r)[ijm(kRR)R̂mkRRJm(kRR)φ̂+kRβJm(kRR)ẑ]eimφeiβz,
(5)

for p polarization by using cylindrical coordinates r⃗=(R,φ,z), m is the azimuthal quantum number, and β=ωv is the momentum along the z axis, kR=(kn(r))2β2 is the in-plane component of the wave vector, k is the wave number of the light in the vacuum, n(r⃗) is the refractive index of the martial. The Eqs. (4) and (5) are defined as standing waves, the outgoing waves EHi,βmσ(r⃑) (σ(σ′)=s, p) can be obtained by substituting the Hankel function H (1) m for the Bessel function Jm in the expressions. With this choice, the outgoing waves vanish in the far-field limit. The coefficients {ψinl,msinl,mp} 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

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]

]:

[ψi,msinψi,mpin]=[mi,ssmi,spmi,psmi,pp]{[mi,ssmi,spmi,psmi,pp]1{[ψi,msψi,mp][mi,ssmi,spmi,psmi,pp][0Qmext]}}.
+[mi,ssmi,spmi,psmi,pp][0Qmext]
(6)

Where i,σσ m and ℑi,σσ′ m are the matrix elements of the external reflection and transmission matrices of the i th cylinder, ℜ′i,σσ′ m and ℑ′i,σσ′ m are the matrix elements of the internal reflection and transmission matrices. The explicit expressions for them are given in Appendix. The Qmext=πkRkεωJm(kRb) is the expanded coefficient of the field radiated by the moving electron [18

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 {ψi,msi,mp} represents the scattered coefficient outside the cylinders, which can be obtained by solving the following self-consistent equation:

ψi,ms=δilml,spQmext+Σj=1jinΣm=Πmmij(mj,ssψj,ms+mj,spψj,mp)
(7)

and

ψi,mp=δilml,ppQmext+Σj=1jinΣm=Πmmij(mj,psψj,ms+mj,ppψj,mp)
(8)

with

Πmmij=Hmm(1)(kRRij)ei(mm)φij.
(9)

Where δil is taken as 1 for i=l and 0 for the otherwise, (Rij,φij) are the local polar coordinates of the vector RijRjRi, Ri is the position of the ith cylinder. Based on Eqs. (1)(9), the EELS for the 2D QCs can be obtained by the numerical calculations.

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.

The calculated results of the EELS as a function of the frequency for the 10-fold and 12-fold QCs with square-triangle tiling structure are plotted in Figs. 1(a) and 1(b), respectively. Here the dielectric constant of the host medium is taken as ε=11.4+0.1i [18

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]

] and the diameter of the air hole is D=0.58a, a is the lattice constant. The velocity of the moving electron beam is v=0.7c 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=0.8a 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

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]

]. In such a case, the spectra are smooth inside the gaps. That is to say, the new resonant peaks do not appear with the increase of the sample size. This is in contrast to the cases of the QCs. It is seen clearly from Figs. 1(a) and 1(b) that the new resonant peaks inside the dip appear with the increase of the sample size. In the following we focus on our discussions on the physical properties of these new resonant peaks.

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).

Such a phenomenon is not sensitive to the change of the impact parameter. Figure 2(a) displays the energy loss spectra as a function of the frequency for the 12-fold QCs with different impact parameter b at v=0.7c. 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 0.55c, 0.7c and 0.78c, 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 (kR=(kn(r))2β2). This is equivalent to the dispersive nature of the modes as has been pointed out in Refs. [17

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]

, 18

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]

].

Corresponding to the energy loss, the coupling of the electron with radiation modes of the system gives rise to radiation emission. The present investigation can be directly applied to cathodoluminescence (CL) in the system. In general, the CL emission probabilities are always smaller than the energy loss probabilities due to the existence of the absorption in the materials. They converge to the same value only in the absence of the absorption [15

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

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]

]. In addition, we would like to point out that the above results are only for one kind of material, if we choose other kinds of material such as Al 2 O 3, similar phenomena can also be found.

3. Local density of states in 2D photonic QCs

In this section we calculate 3D LDOS in 2D photonic QCs by using the multiple-scattering method. Such a method for the periodic PCs has been developed in Ref.[37

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]

]. Here we extend it to the quasi-periodic structures. The three-dimensional local density of states can be expressed by 3D Green tensor as [37

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]

]

ρ(r,ω)=2ωπIm{Tr[GE(r,r,ω)]}.
(10)

Here GE(r⃑,r⃑′,ω) is the electromagnetic Green function with a source at the location r⃗′ and observation point at r⃗, which can be obtained by applying the following inverse Fourier transformation:

GE(r,r,ω)=12πdβeiβ(zz)G˜E(R,R,β,ω),
(11)

where

G˜E(R,R,β,ω)=(G˜xxEG˜xyEG˜xzEG˜yxEG˜yyEG˜yzEG˜zxEG˜zyEG˜zzE).
(12)

Here the matrix element G̃Euo represents the 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

(R2+kR2)G˜zuV(R,R,β,ω)=Duvδ(R,R)
(13)

with

DuE=δzu+iβk2n(r)2·ûandDuH=ẑ·×û.
(14)

where 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 lth cylinder can be expressed as

G˜zul,V(R,R,β,ω)=i4DuV{H0(1)(kRRR)}+Σm=Czu,ml,VJm(kRR)eimϕ.
(15)

Where the coefficients, {Cl,Ezu,m, Cl,Hzu,m}, can be obtained by the scattered coefficients of the system {Bi,Ezu,m, Bi,Hzu,m} [37

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]

]. The scattered coefficients can be obtained by solving the following Rayleigh identities

Bzu,mi,E=δil(Tml,ssQzu,mE+Tml,spQzu,mH)+Σj=1jinΣm=Smmij(Rmj,ssBzu,mj,E+Rmj,spBzu,mj,H)
(16)

and

Bzu,mi,H=δil(Tml,psQzu,mE+Tml,ppQzu,mH)+Σj=1jinΣm=Smmij(Rmj,psBzu,mj,E+Rmj,ppBzu,mj,H)
(17)

with

Smmij=Hmm(1)(kRRij)ei(mm)φij.
(18)

The coefficients {QEzu,m, QHzu,m} are the results of the point source. And R j,σσ′ m and T l,σσ′ m are the matrix elements of the reflection and transmission matrices [37

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]

]. Then, z component of the LDOS is

ρ˜z(R,β,ω)=2ωπIm[ẑ·G˜E(R,R,β,ω)·ẑ]=2ωπIm[G˜zzE(R,R,β,ω)].
(19)

Similarly, the other components of the LDOS can also be obtained.

QCs, it is also found in other high-symmetric QCs. The corresponding results for 10-fold QCs are shown in Figs. 4(a), 4(b) and 4(c). The distribution of the eigen-field (|Ez|) in Fig. 4(c) corresponds to the resonant peak in Fig. 4(a) at ωa/2πc=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.

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. (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).

The above results are only for the case with 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 (ρ˜ =ρ˜ x+ρ˜ y+ρ˜ z). 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/c.

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.

4. Conclusion

Based on the multiple-scattering method, we have investigated the interactions of the moving electron beam with the two-dimensional 10-fold and 12-fold photonic QCs. The EELS and 3D LDOS in these systems have been calculated. Some three-dimensional localized states with certain momentum along the axis of the cylinder in these defect-free two-dimensional systems have been found. The exact corresponding between these localized states and the resonant peaks in the EELS have been demonstrated. This means that the defect-free localized states in these systems can be explored by means of the EELS.

Appendix

Here we provide the explicit expressions for the matrix elements of the external (or internal) reflection and transmission matrices. The matrix elements of the external reflection matrix are

mss=1δm[(k2αH+Jk+2αJH+)(αJJ+αJ+J)m2τ2J+H+J2],
(20)
msp=1δmmτJ22ik+kRπkR+2a0=+ps,
(21)
mpp=1δm[(k2αJ+Jk+2αJJ+)(αJH+αH+J)m2τ2J+H+J2],
(22)
mps=1δmmτJ22ik+kRπkR+2a0.
(23)

The corresponding matrix elements of the external transmission matrix are

mps=1δmmτ2ik+JH+πkRa0,
(24)
mpp=1δm(αJH+αH+J)2ikk+πkR+a0,
(25)
msp=1δmmτ2ikJH+πkRa0,
(26)
mss=1δm(k2αH+Jk+2αJH+)2iπkRa0.
(27)

The matrix elements of the internal reflection matrix are

mss=1δm[(k2αH+Jk+2αJH+)(αHH+αH+H)m2τ2JHH+2],
(28)
msp=1δmmτH+22ikkR+πkR2a0,
(29)
mps=1δmmτH+22ikkR+πkR2a0,
(30)
mpp=1δm[(k2αH+Jk+2αJH+)(αJH+αH+J)m2τ2JHH+2].
(31)

The corresponding matrix elements of the internal transmission matrix are

mps=1δmmτ2ikJH+πkR+a0,
(32)
mpp=1δm(αJH+αH+J)2ikk+πkR+a0,
(33)
msp=1δmmτ2ik+JH+πkR+a0,
(34)
mss=1δm(k2αH+Jk+2αJH+)2iπk+a0.
(35)

Where the k + and k are the wave vectors outside and inside the cylinder, respectively, kR±=k±2β2 represents the in-plane component of the wave vector. a0 is the radius of the cylinder.

τ=(kR2kR+2)βkR+kRa0,
(36)
δm=m2τ2J2H+2+(αH+JαJH+)(k2αJH+k+2αH+J),
(37)
αJH+=JmHm+kR+,
(38)
Jm±=Jm(kR±a0).
(39)

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No.10825416) and the National Key Basic Research Special Foundation of China under Grant 2007CB613205.

References and links

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]

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]

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 35, 7365–7371 (1987). [CrossRef]

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 59, 3095–3107 (1999). [CrossRef]

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]

14.

J. B. Pendry and L. Martin-Moreno, “Energy-loss by charged-particles in complex media,” Phys. Rev. B 50, 5062–5073 (1994). [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]

16.

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]

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]

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]

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]

20.

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

21.

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

22.

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

23.

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

24.

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

25.

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]

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. 73, 1368–1371 (1994). [CrossRef] [PubMed]

27.

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]

28.

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

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 404, 740–743 (2000). [CrossRef] [PubMed]

31.

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]

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. 94, 183903 (2005). [CrossRef] [PubMed]

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]

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]

36.

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

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]

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

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  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]
  14. J. B. Pendry and L. Martın-Moreno, "Energy-loss by charged-particles in complex media," Phys. Rev. B 50, 5062-5073 (1994). [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]
  16. 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]
  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]
  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]
  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]
  20. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal-Molding the Flow of Light, (Princeton University Press, Princeton, NJ, 1995).
  21. C. M. Soukoulis, Photonic Band Gap Materials, (Kluwer, Academic, Dordrecht,1996).
  22. K. Sakoda, Optical properties of photonic crystals, (Springer, 2001).
  23. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  24. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
  25. 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]
  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. 73, 1368-1371 (1994). [CrossRef] [PubMed]
  27. 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]
  28. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997). [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]
  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 404, 740-743 (2000). [CrossRef] [PubMed]
  31. 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]
  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. 94, 183903 (2005). [CrossRef] [PubMed]
  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]
  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]
  36. K. Mnaymneh and R. C. Gauthier, Mode localization and band-gap formation in defect-free photonic quasicrystalsOpt. Express 14, 5089-5099 (2007). [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]
  38. 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]

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