Interaction of fast electron beam with photonic quasicrystals

Wei Zhong; Jinying Xu; Xiangdong Zhang

doi:10.1364/OE.17.013270

1. Introduction

Scanning transmission electron microscopy (STEM) has proved to be a powerful technique for determining different microstructures of nanometer scale [1]. 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 [2–17]. 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, 3], cylinders [4,5], hyperbolic wedges [6], triangles [7], spheres[8–11], spheroid [12] or combination of spherical interfaces and a plane[13]. Recently, the discussions have been extended to more complex geometries such as photonic crystal (PCs) [13–19].

The PCs are regular arrays of materials with different refractive indices, which have been under intensive study during the past two decades [20–24]. Since the PCs can have a spectral gap in which electromagnetic wave propagation is forbidden in all directions, it offers the possibility of controlling the flow of photons in a way analogous to electrons in a semiconductor. It can have profound implications for quantum optics, high-efficiency lasers, optoelectronic devices, and other areas of applications [25–28, 20–24]. Recent investigations have shown that the band structure and density of states of the two-dimensional (2D) PCs are directly related to the EELS [18, 19]. The band-gap properties of the PCs can be probed by the EELS. However, all these investigations focus on the period structures.

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 [29–33]. 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 [34–36]. 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

We consider a 2D photonic quasicrystal consisting of infinitely long parallel cylindrical air cavities drilled in a host medium. When an electron moves along a straight-line trajectory parallel to the axis of cylinder inside the air cavity, the radiation field may engender collective excitations of the medium that act back upon the electron, which leads to the energy loss of the moving electron. The energy loss can be related to the force exerted by the induced electric field E⃗^ind acting on it as [10, 11, 18]

Δ E = \int d t \overset{⇀}{v} \cdot {\overset{⇀}{E}}^{ind} ({\overset{⇀}{r}}_{t}, t) = L \int_{0}^{\infty} ω d ω P (ω),

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

P (ω) = \frac{1}{π ω L} \int dt Re {e^{- i ω t} \overset{⇀}{v} \cdot {\overset{⇀}{E}}^{ind} ({\overset{⇀}{r}}_{t}, ω)} .

represents the electron energy loss probability (EELS) per unit of path length in frequency space ω. Here E⃑^ind 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

{\overset{⇀}{E}}^{ind} (\overset{⇀}{r}, ω) = \underset{m}{Σ} (ψ_{l, ms}^{in} E_{l, β ms}^{J} (\overset{⇀}{r}) + ψ_{l, mp}^{in} E_{l, β mp}^{J} (\overset{⇀}{r})),

where

E_{i, β ms}^{J} (\overset{⇀}{r}) = [\frac{im}{k_{R} R} J_{m} (k_{R} R) \hat{R} - {J'}_{m} (k_{R} R) \hat{φ}] e^{im φ} e^{i β z},

for s polarization, and

E_{l, β mp}^{J} (\overset{⇀}{r}) = \frac{β}{kn (\overset{⇀}{r})} [i {j'}_{m} (k_{R} R) \hat{R} - \frac{m}{k_{R} R} J_{m} (k_{R} R) \hat{φ} + \frac{k_{R}}{β} J_{m} (k_{R} R) \hat{z}] e^{im φ} e^{i β z},

for p polarization by using cylindrical coordinates r⃗=(R,φ,z), m is the azimuthal quantum number, and $β = ω ⁄ v$ is the momentum along the z axis, $k_{R} = \sqrt{{(kn (r))}^{2} - β^{2}}$ 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 E^H_i,_βmσ(r⃑) (σ(σ′)=s, p) can be obtained by substituting the Hankel function H ⁽¹⁾ _m for the Bessel function J_m in the expressions. With this choice, the outgoing waves vanish in the far-field limit. The coefficients {ψⁱⁿ_l,ms,ψⁱⁿ_l,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]:

[\begin{matrix} ψ_{i, ms}^{in} \\ ψ_{i, mp}^{in} \end{matrix}] = [\begin{matrix} {ℜ'}_{m}^{i, ss} & {ℜ'}_{m}^{i, sp} \\ {ℜ'}_{m}^{i, ps} & {ℜ'}_{m}^{i, pp} \end{matrix}] {{[\begin{matrix} ℜ_{m}^{i, ss} & ℜ_{m}^{i, sp} \\ ℜ_{m}^{i, ps} & ℜ_{m}^{i, pp} \end{matrix}]}^{- 1} {[\begin{matrix} ψ_{i, ms} \\ ψ_{i, mp} \end{matrix}] - [\begin{matrix} ℑ_{m}^{i, ss} & ℑ_{m}^{i, sp} \\ ℑ_{m}^{i, ps} & ℑ_{m}^{i, pp} \end{matrix}] [\begin{matrix} 0 \\ Q_{m}^{ext} \end{matrix}]}} .

Where ℜ ^i,σσ′ _m and ℑ^i,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 $Q_{m}^{ext} = \frac{π k_{R} k}{ε ω} J_{m} (k_{R} b)$ is the expanded coefficient of the field radiated by the moving electron [18], ε is the relative permittivity. Here {ψ^i,ms,ψ^i,mp} represents the scattered coefficient outside the cylinders, which can be obtained by solving the following self-consistent equation:

ψ_{i, ms} = δ_{il} ℑ_{m}^{l, sp} Q_{m}^{ext} + Σ_{\underset{j \neq i}{j = 1}}^{n} Σ_{m' = - \infty}^{\infty} Π_{mm'}^{ij} (ℜ_{m'}^{j, ss} ψ_{j, ms} + ℜ_{m'}^{j, sp} ψ_{j, mp})

and

ψ_{i, mp} = δ_{il} ℑ_{m}^{l, pp} Q_{m}^{ext} + Σ_{\underset{j \neq i}{j = 1}}^{n} Σ_{m' = - \infty}^{\infty} Π_{mm'}^{ij} (ℜ_{m'}^{j, ps} ψ_{j, ms} + ℜ_{m'}^{j, pp} ψ_{j, mp})

with

Π_{mm'}^{ij} = H_{m - m'}^{(1)} (k_{R} R_{ij}) e^{i (m' - m) φ_{ij}} .

Where δ_il is taken as 1 for i=l and 0 for the otherwise, (R_ij,φ_ij) are the local polar coordinates of the vector R⃑_ij≡R⃑_j−R⃑_i, R⃑_i 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.

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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] 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]. 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).

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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 $(k_{R} = \sqrt{{(kn (r))}^{2} - β^{2}})$ . This is equivalent to the dispersive nature of the modes as has been pointed out in Refs. [17, 18].

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, 38]. 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.

Why do the EELS for the QCs appear some different features in comparison with that of the periodic structure? What do the new peaks in the EELS mean? In order to answer these questions, we calculate the three-dimensional (3D) LDOS of the system in the next section.

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]. 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]

ρ (\overset{⇀}{r}, ω) = - \frac{2 ω}{π} Im {Tr [G^{E} (\overset{⇀}{r}, \overset{⇀}{r}, ω)]} .

Here G^E(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:

G^{E} (\overset{⇀}{r}, \overset{⇀}{r'}, ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} d β e^{i β (z - z')} {\tilde{G}}^{E} (\overset{⇀}{R}, \overset{⇀}{R'}, β, ω),

where

{\tilde{G}}^{E} (\overset{⇀}{R}, \overset{⇀}{R}', β, ω) = (\begin{matrix} {\tilde{G}}_{xx}^{E} & {\tilde{G}}_{xy}^{E} & {\tilde{G}}_{xz}^{E} \\ {\tilde{G}}_{yx}^{E} & {\tilde{G}}_{yy}^{E} & {\tilde{G}}_{yz}^{E} \\ {\tilde{G}}_{zx}^{E} & {\tilde{G}}_{zy}^{E} & {\tilde{G}}_{zz}^{E} \end{matrix}) .

Here the matrix element G̃^E_uo 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

(\nabla_{R}^{2} + k_{R}^{2}) {\tilde{G}}_{zu}^{V} (\overset{⇀}{R}, \overset{⇀}{R}', β, ω) = D_{u}^{v} δ (\overset{⇀}{R}, \overset{⇀}{R'})

with

D_{u}^{E} = δ_{zu} + \frac{i β}{k^{2} n {(\overset{⇀}{r})}^{2}} \nabla \cdot \hat{u} and D_{u}^{H} = \hat{z} \cdot \nabla \times \hat{u} .

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

{\tilde{G}}_{zu}^{l, V} (\overset{⇀}{R}, \overset{⇀}{R}', β, ω) = - \frac{i}{4} D_{u}^{V} {H_{0}^{(1)} (k_{R} ∣ \overset{⇀}{R} - \overset{⇀}{R'} ∣)} + Σ_{m = - \infty}^{\infty} C_{zu, m}^{l, V} J_{m} (k_{R} R) e^{im ϕ} .

Where the coefficients, {C^l,E_zu,m, C^l,H_zu,m}, can be obtained by the scattered coefficients of the system {B^i,E_zu,m, B^i,H_zu,m} [37]. The scattered coefficients can be obtained by solving the following Rayleigh identities

B_{zu, m}^{i, E} = δ_{il} (T_{m}^{l, ss} Q_{zu, m}^{E} + T_{m}^{l, sp} Q_{zu, m}^{H}) + Σ_{\underset{j \neq i}{j = 1}}^{n} Σ_{m' = - \infty}^{\infty} S_{mm'}^{ij} (R_{m'}^{j, ss} B_{zu, m'}^{j, E} + R_{m'}^{j, sp} B_{zu, m'}^{j, H})

and

B_{zu, m}^{i, H} = δ_{il} (T_{m}^{l, ps} Q_{zu, m}^{E} + T_{m}^{l, pp} Q_{zu, m}^{H}) + Σ_{\underset{j \neq i}{j = 1}}^{n} Σ_{m' = - \infty}^{\infty} S_{mm'}^{ij} (R_{m'}^{j, ps} B_{zu, m'}^{j, E} + R_{m'}^{j, pp} B_{zu, m'}^{j, H})

with

S_{mm'}^{ij} = H_{m - m'}^{(1)} (k_{R} R_{ij}) e^{i (m' - m) φ_{ij}} .

The coefficients {Q^E_zu,m, Q^H_zu,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]. Then, z component of the LDOS is

{\tilde{ρ}}_{z} (\overset{⇀}{R}, β, ω) = - \frac{2 ω}{π} Im [\hat{z} \cdot {\tilde{G}}^{E} (\overset{⇀}{R}, \overset{⇀}{R}, β, ω) \cdot \hat{z}] = - \frac{2 ω}{π} Im [{\tilde{G}}_{zz}^{E} (\overset{⇀}{R}, \overset{⇀}{R}, β, ω)] .

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

The dashed lines in Figs. 3(a) and 3(b) represent the calculated results of z-projected LDOS $\tilde{ρ}$ _z as a function of the frequency for 12-fold QCs (triangular-square tiling structure) with different size and electron velocity, respectively. In order to provide comparison, the corresponding EELS are also plotted as solid lines in the figures. Comparing them, we find that the corresponding between them is very well. Due to the effect of the dispersive nature on the LDOS, some changes appear for the position and number of the peaks with the change of the electron velocity. Such changes in the LDOS are exactly equivalent to those in the EELS. This is because the energy loss probability is directly related to the LDOS, $P (\overset{⇀}{R}, ω) = \frac{2 π e^{2} L}{ℏ ω} {\tilde{ρ}}_{z} (\overset{⇀}{R}, β, ω)$ , which has been pointed out in Ref.[38]. This means that the physical intension of the resonant peak in the EELS can be understood from the corresponding peak in the LDOS.

The previous discussions on 2D LDOS have shown that 2D localized states exist in some 2D photonic QCs even though their patterns can be defect-free [34–36]. These localized states can be determined by the resonant peaks in the 2D LDOS. Here our 3D LDOS calculations demonstrate that some 3D localized states with certain β can also be observed in the 2D photonic QCs, which some of them has been marked by arrows in Fig. 3(a). In order to demonstrate them further more, the distributions of the eigen-field (|E_z|) in the sample are calculated. The calculated result at ωa/2πc=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

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 (|E_z|) 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 $\tilde{ρ}$ _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) $\tilde{ρ}$ _z. (b) The corresponding result with different electron velocities at N=55. (c) The distribution of eigen-electric field (|E_z|) inside the sample with N=55 at ωa/2πc=0.318. The field (E_z) map is in linear false-color scale (red=high; blue=low). The other parameters are identical with those in Fig. 1 (b).

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Fig. 4. (a). Comparison between EELS P(ω) and LDOS $\tilde{ρ}$ _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 (|E_z|) inside the sample with N=141 at ωa/2πc=0.324. The field (|E_z|) map is in linear false-color scale (red=high; blue=low). The other parameters are identical with those in Fig. 1(a).

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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 ( $\tilde{ρ}$ = $\tilde{ρ}$ _x+ $\tilde{ρ}$ _y+ $\tilde{ρ}$ _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) $\tilde{ρ}$ _z; (c) unprojected LDOS (−π/2ω) $\tilde{ρ}$ . The other parameters are the same to those in Fig. 3.

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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

ℜ_{m}^{ss} = \frac{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}],

ℜ_{m}^{sp} = \frac{1}{δ_{m}} m τ J^{- 2} \frac{2 {ik}^{+} k_{R}^{-}}{π k_{R}^{+ 2} a_{0}} = ℜ^{+ ps},

ℜ_{m}^{pp} = \frac{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}],

ℜ_{m}^{ps} = \frac{1}{δ_{m}} m τ J^{- 2} \frac{2 i k^{+} k_{R}^{-}}{π k_{R}^{+ 2} a_{0}} .

The corresponding matrix elements of the external transmission matrix are

ℑ_{m}^{ps} = \frac{1}{δ_{m}} m τ \frac{2 i k^{+} J^{-} H^{+}}{π k_{R}^{-} a_{0}},

ℑ_{m}^{pp} = \frac{1}{δ_{m}} (α_{J^{-} H^{+}} - α_{H^{+} J^{-}}) \frac{2 i k^{-} k^{+}}{π k_{R}^{+} a_{0}},

ℑ_{m}^{sp} = \frac{1}{δ_{m}} m τ \frac{2 i k^{-} J^{-} H^{+}}{π k_{R}^{-} a_{0}},

ℑ_{m}^{ss} = \frac{1}{δ_{m}} (k^{- 2} α_{H^{+} J^{-}} - k^{+ 2} α_{J^{-} H^{+}}) \frac{2 i}{π k_{R}^{-} a_{0}} .

The matrix elements of the internal reflection matrix are

{ℜ'}_{m}^{ss} = \frac{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}],

{ℜ'}_{m}^{sp} = \frac{1}{δ_{m}} m τ H^{+ 2} \frac{2 i k^{-} k_{R}^{+}}{π k_{R}^{- 2} a_{0}},

{ℜ'}_{m}^{ps} = \frac{1}{δ_{m}} m τ H^{+ 2} \frac{2 i k^{-} k_{R}^{+}}{π k_{R}^{- 2} a_{0}},

{ℜ'}_{m}^{pp} = \frac{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}] .

The corresponding matrix elements of the internal transmission matrix are

{ℑ'}_{m}^{ps} = \frac{1}{δ_{m}} m τ \frac{2 i k^{-} J^{-} H^{+}}{π k_{R}^{+} a_{0}},

{ℑ'}_{m}^{pp} = \frac{1}{δ_{m}} (α_{J^{-} H^{+}} - α_{H^{+} J^{-}}) \frac{2 i k^{-} k^{+}}{π k_{R}^{+} a_{0}},

{ℑ'}_{m}^{sp} = \frac{1}{δ_{m}} m τ \frac{2 i k^{+} J^{-} H^{+}}{π k_{R}^{+} a_{0}},

{ℑ'}_{m}^{ss} = \frac{1}{δ_{m}} (k^{- 2} α_{H^{+} J^{-}} - k^{+ 2} α_{J^{-} H^{+}}) \frac{2 i}{π k^{+} a_{0}} .

Where the k ⁺ and k ⁻ are the wave vectors outside and inside the cylinder, respectively, $k_{R}^{\pm} = \sqrt{k^{\pm 2} - β^{2}}$ represents the in-plane component of the wave vector. a0 is the radius of the cylinder.

τ = \frac{(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}^{\pm} = J_{m} (k_{R}^{\pm} a_{0}) .

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]

Interaction of fast electron beam with photonic quasicrystals

Abstract

1. Introduction

2. Electron energy loss spectroscopy in 2D photonic QCs

3. Local density of states in 2D photonic QCs

4. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (5)

Equations (40)

Optics Express