Electron energy loss and Smith-Purcell radiation in two- and three-dimensional photonic crystals

Tetsuyuki Ochiai; Kazuo Ohtaka

doi:10.1364/OPEX.13.007683

1. Introduction

When a moving charged particle (e.g. electron) passes near a specimen, the particle loses its kinetic energy via the electro-magnetic interaction between the particle and the specimen. The total amount of the loss, which is called electron energy loss (EEL), reflects significantly the bulk and surface excitation of the specimen. Recent investigation on the EEL in photonic crystals (PC’s) have shown that the loss spectrum can serve as a probe of them [1, 2, 3, 4, 5, 6]. We can probe bulk photonic modes, photonic band gap and pseudo gap, localized defect mode, surface mode, etc. with the EEL spectrum. In addition, under certain conditions, a radiation emission, so-called Smith-Purcell radiation (SPR) [7], is induced from PC. This radiation is scalable in frequency, and thus can be used as a novel radiation source without limitation of frequency.

Here, we present a theoretical analysis of the EEL and SPR spectra in a two-dimensional(2d) PC slab [8] composed of a periodic array of air holes fabricated in a thin dielectric slab, and in a three-dimensional(3d) woodpile PC [9] composed of high-index rectangular stripes. The method employed in this study is the S-matrix one in terms of eigenmode expansion on the basis of plane waves [10, 11, 12, 13]. The method is quite powerful as long as the system concerned can be regarded as a finite stack of one- or two-dimensional gratings which are homogeneous along the stacking direction. This is the case for the 2d PC slab and the 3d woodpile PC.

So far, studies on SPR are limited mainly in metallic diffraction gratings [14, 15, 16, 17, 18, 19, 20], periodic arrays of infinitely-long cylinders [1, 5, 6], and periodic array of spheres [1, 21, 22, 23, 24]. However, to the best of our knowledge, none has reported on the EEL and SPR in the 2d PC slabs and the 3d woodpile PC’s. Compared to the systems studied so far, these PC’s are quite efficient in various applications of optoelectronics. The 2d PC slab is generally favorable for realizing photonic integrated circuits, because sophisticated techniques for thin-film formation and lithography are available. The 3d woodpile PC has the wide complete photonic band gap between the second and third bands, and thus the gap is robust against disorder. Therefore, the woodpile can be a host of various artificial defects [25, 26], in which novel functions can be realized through localized defect modes. Since these PC’s are joints favorites for various applications, it is quite important to integrate knowledge of them from different points of view. In this paper we focus on the interaction between a moving charged particle and these PC’s from the point of view of probe and novel radiation source.

In this paper we also present the EEL and SPR when the charged particle moves inside the woodpile PC. This case is related to the Cherenkov effect of PC’s reported previously[3, 4, 27]. There, owing to the peculiar dispersion relation of photon inside the PC’s, the charged particle moving inside the PC’s can give rise to a backward-pointing Cherenkov radiation inside the PC’s. In addition, the EEL spectrum through a PC is affected strongly by such a novel Cherenkov radiation. In particular, the authors of Ref. [3, 4] showed that the photonic band structure of a certain kind of PC’s can be directly mapped by the angle-resolved EEL spectrum. Though such a direct-mapping is not possible in our case, it is certainly true that the EEL and SPR spectra can probe the photonic band structures of the 2d PC slab and the 3d woodpile PC.

This paper is organized as follows. A theoretical formalism to deal with the EEL and SPR in these PC’s is presented in Sec. 2. Secs. 3 and 4 are devoted to present numerical results of the EEL and SPR, discussing their relevance to the photonic band structures. Finally, we summarize the results.

2. Formalism

Let us consider a charged particle moving below a 2d PC slab composed of a square array of air holes with radius r embedded in a thin dielectric slab of thickness d and dielectric constant ε. A schematic illustration of the system under study is shown in Fig. 1. For simplicity, the charged particle (charge e) is assumed to move with constant velocity v along the (1,0) direction of the square lattice. The trajectory of the particle is given by x _t = (vt,y ₀,z ₀) in the Cartesian coordinate. Here, the planes of z = 0 and y = 0 are taken to the mirror plane bisecting the slab and the mirror plane bisecting a column of air cylinders, respectively.

The moving charged particle is accompanied by the evanescent radiation whose electric field is given by

E (x; ω) = \int_{- \infty}^{\infty} \frac{{dk}_{y}}{2 π} ε^{\pm} (\frac{ω}{v}, k_{y}) e^{{i K}^{\pm} ∙ (x - x_{0})},

ε^{\pm} (\frac{ω}{v}, k_{y}) = - \frac{μ_{0} eω}{2} (\begin{matrix} \frac{1}{Γ} (1 - \frac{1}{q^{2}} {(\frac{ω}{v})}^{2}) \\ - \frac{1}{{Γ q}^{2}} \frac{ω}{v} k_{y} \\ \mp \frac{1}{q^{2}} \frac{ω}{v} \end{matrix}),

Fig. 1. A schematic illustration of the two-dimensional (2d) photonic crystal (PC) slab under study. It consists of a square array (lattice constant a) of cylindrical air holes with radius r, embedded in a thin dielectric slab with thickness d and dielectric constant ε. A charged particle of charge e moves with constant velocity v below the slab, along the (1,0) direction of the square lattice.

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K^{\pm} = (\frac{ω}{v}, k_{y}, ±Γ), Γ = \sqrt{q^{2} - {(\frac{ω}{v})}^{2} - k_{y}^{2}} q = \frac{ω}{c},

in the time-Fourier component. Here, the superscript of ε refers to the sign of z - z ₀. It should be noted that the wave number in the x direction is fixed as ω/v and that the radiation field is given by the integral over k_y . Since the velocity of the particle is less than c, Γ is pure imaginary irrespective of k_y . In the following discussion we call the dispersion line ω = vk_x the v-line. As can be easily checked, the radiation field is symmetric under the y(z)-inversion with respect to y ₀(z ₀). Thus, when the trajectory lies in a mirror plane of the PC slab, solely the symmetric modes with respect to the plane can be excited. Of course, this selection rule is broken otherwise.

This evanescent radiation is scattered by the PC slab and the scattered wave can turn into a propagating radiation by acquiring an Umklapp momentum transfer from the PC slab. This scattering process is described with the S-matrix of the PC slab, which is obtained numerically with the eigenmode expansion method on the basis of plane waves [10, 11, 12, 13]. After the scattering, the transmitted radiation through the PC slab becomes

E^{tra} (x; ω) = \sum_{G} \int_{- \frac{π}{a}}^{\frac{π}{a}} \frac{{dk}_{y}}{2 π} e^{i K_{G}^{+} ∙ x} t_{G}^{\pm} (ω, k_{y}),

t_{G}^{+} (ω, k_{y}) = \sum_{G'} S_{+ +} (G, G') δ_{{G'}_{x}, G_{x}^{0}} ε_{G'}^{+} e^{- i k_{G'}^{+} ∙ x_{0}},

ε_{G}^{\pm} = ε^{\pm} (k_{x} + G_{x}, k_{y} + G_{y}),

K_{G}^{\pm} = (k_{x} + G_{x}, k_{y} + G_{y} \pm Γ_{G}),

Γ_{G} = \sqrt{q^{2} - {(k_{x} + G_{x})}^{2} - {(k_{y} + G_{y})}^{2} .}

Here, G is a reciprocal lattice vector of the square lattice, S _±± (G, G′) is the S-matrix, and k_x is defined in such a way that ω/v = k_x + $G_{x}^{0}$ , being k_x inside the first Brillouin zone. In a similar manner the reflected radiation becomes

E^{ref} (x; ω) = \sum_{G} \int_{- \frac{π}{a}}^{\frac{π}{a}} \frac{{dk}_{y}}{2 π} e^{i K_{G}^{-} ∙ x} t_{G}^{-} (ω, k_{y}),

t_{G}^{-} (ω, k_{y}) = \sum_{G'} S_{- +} (G, G') δ_{G_{x}^{'}, G_{x}^{0}} ε_{G'}^{+} e^{- i K_{G'}^{+} ∙ x_{0}} .

The change of the wave vectors in the integrand of Eqs.(4) and (9) is due to the Umklapp shift.

The electron energy loss spectrum is defined by the work done by the exerted force reacting back to the charged particle (electron). The work W_el per unit length of the trajectory is given by

W_{el} = \frac{2}{π} \int_{0}^{\infty} dω P_{pel} (ω),

P_{el} (ω) = - \frac{e}{2 L} Re [\int {dte}^{- iωt} v ∙ E^{ref} (x_{t}; ω)],

= \int_{- \frac{π}{a}}^{\frac{π}{a}} \frac{{dk}_{y}}{2 π} P_{el} (ω, k_{y}),

P_{el} (ω, k_{y}) = \frac{- e}{2} \sum_{G_{y}, G_{y}^{'}} Re [e^{i (G_{y} - G_{y}^{'}) y_{0} - i (Γ_{G_{X}^{0}, G_{y}} + Γ_{G_{x}^{0}, G_{y}^{'}}) z_{0}} (S_{- +} (G_{x}^{0} G_{y}; G_{x}^{0} G_{y}^{'}) ε_{G_{x}^{0}, G_{y}^{'}}^{+})_{x}]

Here, L is the total length of the trajectory. On the other hand, the SPR spectrum is defined by the total radiation energy emitted from the PC. The radiation energy W_sp per unit length of the trajectory is given by

W_{sp} = \frac{2}{π} \int_{0}^{\infty} dω P_{sp} (ω),

P_{sp} (ω) = \frac{1}{a} \int_{0}^{a} dx \int_{- \infty}^{\infty} dy \frac{1}{2} Re {(E^{*} (x; ω) \times H (x; ω))}_{z}

= \int_{- \frac{π}{a}}^{\frac{π}{a}} \frac{{dk}_{y}}{2 π} P_{sp} (ω, k_{y}),

P_{sp} (ω, k_{y}) = \frac{1}{{2 μ}_{0} ω} \sum_{G \in open} Γ_{G} ({∣ t_{G}^{+} ∣}^{2} + {∣ t_{G}^{-} ∣}^{2}),

where the summation over G is limited in the open diffraction channels. If the PC slab has no absorption loss (i.e., Imε = 0), the above two spectra must coincide, that is, P_el (ω, k_y ) = P_sp (ω, k_y ), as can be verified by considering the flux conservation through the PC slab. Since the SPR has the infrared cutoff, the EEL is exactly zero below the cutoff frequency. However, by introducing absorption loss, the EEL has finite value below the cutoff frequency, reflecting the band structure outside the light cone.

The above procedure for the 2d PC slab can be easily generalized to a certain class of 3d PC’s which can be regarded as a stack of the 2d PC slabs having the same periodicity in plane. This is the case for a 3d woodpile PC, which is a promising structure having the wide omnidirectional photonic band gap. As shown in Fig. 2, a 3d woodpile PC consists of a stack of alternating lamellar gratings with orthogonal orientations and the same lattice constant. The S-matrix of each lamellar grating can be obtained in terms of eigenmode expansion on the basis of plane waves, assuming it has a translational symmetry of the square lattice in plane. By the layer doubling procedure [28], it is possible to construct the S-matrix of the whole structure. Moreover, a planar defect of removing a mono-layer of the stripes is readily incorporated in the S-matrix method [29]. Then, using Eq.(14) and (18), we can obtain the EEL and SPR spectra of the 3d woodpile PC’s with and without the planar defect.

So far, we have considered the particle trajectory lying outside the PC’s. However, in the case of the woodpile, the particle can move inside the PC’s. This is indeed possible by taking the particle trajectory to be parallel to the grooves of a x-oriented lamellar grating. In order to adapt the S-matrix method to this case, we need to introduce a virtual homogeneous layer having infinitely small thickness embedded in the stripe layer including the particle trajectory. The particle is supposed to move in the virtual layer. In the virtual layer, the induced electric field turns out to be

Fig. 2. A schematic illustration of the three-dimensional (3d) woodpile PC under study. It consists of a stack of alternating lamellar gratings with orthogonal orientations and the same lattice constant a. The rectangular stripes in the gratings have width w, thickness d, and dielectric constant ε.

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E^{ind} (x; ω) = \sum_{G} \int_{- \frac{π}{a}}^{\frac{π}{a}} \frac{{dk}_{y}}{2 π} (e^{{i K}_{G}^{+} ∙ (x - x_{0})} a_{G}^{+} (ω, k_{y}) + e^{{i K}_{G}^{-} ∙ (x - x_{0})} a_{G}^{-} (ω, k_{y})),

a_{G}^{+} = {[{{(1 - S}_{+ -}^{l} S_{- +}^{u})}^{- 1} S_{+ -}^{l} (ε^{-} + {S_{- +}^{u} ε^{+}}^{})]}_{G},

a_{G}^{-} = {[{{(1 - S}_{- +}^{u} S_{+ -}^{l})}^{- 1} S_{- +}^{u} (ε^{+} + {S_{+ -}^{l} ε^{-}}^{})]}_{G},

where $S_{\pm \pm}^{u (l)}$ is the S-matrix of the upper(lower) layers above(below) the virtual layer and the summation over reciprocal lattice vectors and tensor indices is implicitly assumed in Eqs.(20) and (21). On the other hand, the upper and lower transmitted fields denoted by E ^tra,+ and E ^tra,- respectively are given by

E^{tra, \pm} (x; ω) = \sum_{G} \int_{- \frac{π}{a}}^{\frac{π}{a}} \frac{{dk}_{y}}{2 π} e^{{i K}_{G}^{\pm} ∙ x} t_{G}^{\pm} (ω, k_{y}),

t_{G}^{+} = {[S_{++}^{u} (ε^{+} + a^{+})]}_{G},

t_{G}^{-} = {[S_{- -}^{l} (ε^{-} + a^{-})]}_{G} .

The EEL spectrum thereby becomes

P_{el} (ω, k_{y}) = - \frac{e}{2} \sum_{G_{y}} Re (a_{G_{x}^{0}, G_{y}}^{+} + a_{G_{x}^{0}, G_{y}}^{-}) x .

The above procedure can be easily generalized to the charged particle moving in the planar defect of the woodpile. In this case we do not need to introduce the virtual homogeneous layer. Instead, the planar defect can serve as a virtual layer. Therefore, the same procedure is used to calculate the EEL and SPR spectra.

The spectra reveal a resonance when the Umklapp shifted v-line ω = v(k_x + G_x ) intersects the photonic dispersion curves of the bulk PC in the frequency-momentum space. This is caused by exciting the photonic band mode at the intersection point. A sequence of the intersection points appear at a given velocity of the charged particle, yielding peaks of the EEL and SPR spectra as a function of ω and k_y . By scanning the velocity of the particle, the position of a peak shifts in the frequency-momentum space, tracing a photonic dispersion curve. Therefore, the EEL and SPR spectra can be used as a probe of the photonic band structure.

Fig. 3. The polar coordinate for the Smith-Purcell radiation (SPR). The polar angle θ is defined with respect to the moving direction of the charged particle. The azimuthal angle ϕ is defined on the plane normal to the moving direction.

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From an experimental point of view, the EEL spectrum P_el (ω, k_y ) can be directly measured, because it is proportional to the probability of the charged particle losing h̄ω in its kinetic energy and losing momentum k_y along the y-direction. As for the SPR, its spectrum P_sp (ω,k_y ) is related to the far-field pattern of the radiation, in other words, the frequency-resolved differential cross section of the SPR. Let us introduce the polar coordinate for the SPR as in Fig. 3. Since the output radiation is a superposition of plane waves having wave vector $K_{G}^{\pm}$ , the observed radiation at a given solid angle (θ,ϕ) must have

K_{G}^{\pm} = q (\cos θ, \sin θ \cos ϕ, \sin θ \sin ϕ) .

Thus, k_x ,k_y ,G_x , and G_y involved are uniquely determined as a function of ω, θ, and ϕ. We should note that the polar angle θ has only discrete degree of freedom at a given frequency. Thus, the frequency-resolved cross section of the SPR behaves as Dirac’s delta-function regarding θ. As for the dependence of the azimuthal angle ϕ, the frequency-resolved differential cross section becomes

\frac{\partial^{2} W_{sp}}{\partial ω \partial ϕ} \propto {(\sin θ \cos ϕ)}^{2} {∣ t_{G}^{\pm} (ω, k_{y}) ∣}^{2},

where the superscript of t refers to either transmitted (+) or reflected (-) SPR.

3. Two-dimensional photonic crystal slab

Let us consider the case of a 2d square PC slab. Numerical calculations were performed under the following parameters: r = 0.2a,d = 0.5a,ε = 12 + 0.01i, and z ₀ = -0.5a. The imaginary part of ε was introduced just for representing the absorption loss of the PC slab. Physics of the EEL is unchanged unless the imaginary part is exactly equal to zero. It should be noted that ε can be frequency-dependent, because the method employed in this paper is a “on-shell” one.

First, we show that how the EEL spectrum changes when the hole array is introduced in a thin dielectric slab. In a homogeneous dielectric slab photonic eigenstates are classified into guided modes and radiation modes. The guided modes are confined in the slab and have discrete spectrum outside the light cone, at a given wave number parallel to the slab. On the other hand, the radiation modes are extended in all spatial directions and have continuous spectrum inside the light cone. When a hole array is introduced in the slab, the zone-folding of the guided modes in the momentum space yields quasi-guided modes inside the light cone, which have finite lifetimes owing to the mixing with the radiation modes [30, 31]. The quasi-guided modes, obtained in this way, has a two-dimensional band structure ω = ω_n (k_x ,k_y ). They are connected with the true-guided modes outside the light cone and are embedded in continuous-level radiation modes inside the light cone. Thus, the Fano resonance takes place for the incident light at the frequency of a quasi-guided mode [11]. Moreover, these modes cause a resonance of the EEL and SPR when the shifted v-line intersects the dispersion curves of the true- and quasi-guided modes.

Fig. 4. The electron energy loss (EEL) spectra in the PC slab and in the corresponding homogeneous slab whose dielectric constant is the spatial average of the dielectric function of the PC slab. The resonant frequencies at the intersection points between the (shifted) v-lines and the photonic dispersion curves of Fig. 5 are indicated by arrows. The following parameters were used: ε = 12 + 0.01i,d = 0.5a,r = 0.2a,v = 0.5c,y ₀ = 0.25a, and z ₀ = -0.5a.

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These features are clearly observed when we compare the EEL spectrum in the PC slab with that in the homogeneous dielectric slab whose dielectric constant is the spatial average of the dielectric function in the PC slab. The EEL spectrum in the homogeneous slab is analytically obtained by

P_{el} (ω, k_{y}) = - \frac{e}{2} Re [e^{- 2 i Γ z_{0}} {({Rε}^{+})}_{x}],

where R is the (tensor) reflection coefficient of the homogeneous slab. Fig. 4 shows the EEL spectra P_el (ω, k_y ) (k_y = 0) of both the PC slab and homogeneous slab, assuming v = 0.5c,y0 = 0.25a and z ₀ = -0.5a for the moving charged particle. At low frequencies the EEL spectrum of the PC slab is very close to that of the homogeneous slab, showing the validity of the effective medium approximation. The EEL spectrum of the homogeneous slab has only one peak in the frequency range concerned, whereas that of the PC slab has a sequence of peaks. The peaks of the PC slab are quite sharp, except for the peak whose frequency is very close to that of the homogeneous slab. In order to understand these features we have calculated the photonic band structure of the true- and quasi-guide modes in the PC slab. The band structure can be calculated with the S-matrix of the PC slab. The true-guided modes were obtained from the pole of the S-matrix, whereas the quasi-guided modes were obtained from the peaks of the optical density of states as a function of frequency and momentum parallel to the slab [32].

Fig. 5. The photonic band structure of the two-dimensional square PC slab with the following parameters (See Fig. 1): r = 0.2a,d = 0.5a, and ε = 12. k_y = 0 was assumed. The photonic band modes are classified according to the parities of the y and z inversions. The light line and the (shifted) v-lines of v = 0.5c are indicated by solid and dashed lines, respectively. The mode indicated by arrow is that of the E representation at ωa/2πc = 0.4127.

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Fig. 5 shows the photonic band structure in the PC slab at k_y = 0. The photonic band modes are classified according to the parities of the y and z inversions. As can be clearly seen, the peaks of the EEL spectrum correspond to the intersection points (indicated by arrows in Fig. 4) between the shifted v-lines and the dispersion curves of the true- and quasi-guided modes. The width of each peak is given by the quality factor of the relevant eigenmode and differs largely from mode to mode. As for the peaks of a true-guided mode, the finite widths come from the absorption loss due to the finite imaginary part of ε. If the imaginary part is exactly zero, the quality factors of the true-guided modes have to be infinite, and thus we can not detect the energy loss due to the true-guided mode. On the other hand, a quasi-guided mode has the finite quality factor even if there is no absorption loss. Thus, the energy loss due to the quasi-guide mode is detectable regardless of Imε.

We should recall that the above result was obtained by taking the particle trajectory to be y ₀ = 0.25a, so that the mirror symmetry of the system with respect to the y-coordinate is broken. If the trajectory lies in a mirror plane of the PC slab, a selective excitation of the photonic eigenstates takes place. This is due to the selection rule concerning the parity of the mirror symmetry as was mentioned in the previous section. Fig. 6 shows the y ₀ -dependence of the EEL spectrum in the PC slab. If y ₀ = 0 or y ₀ = 0.5a, the trajectory lies in a mirror plane of the PC slab. Therefore, the y-odd modes with respect to the planes can not be excited, taking account that the incident evanescent radiation has the even parity with respect to the plane including the trajectory. On the other hand, if y ₀ = 0.25a, the trajectory does not lie in the mirror plane, so that such a selection rule is absent. We found that some of the modes (ωa/2πc = 0.3825 and 0.4175) couple very weekly to the particle moving along y ₀ = 0, yielding very small peaks in the EEL spectrum. We suppose a small overlap of the incident evanescent radiation and the relevant photonic eigenstate causes such a small coupling.

So far, we have concentrated on the spectrum of k_y = 0. This is reasonable, because the contribution of k_y = 0 is dominant as can be understood from the role of the exponential factor of Eq.(14). Since Γ_{$G_{x}^{0}$ , G_y} is pure imaginary regardless of k_y and G_y , the exponential factor damps very rapidly with increasing |k_y + G_y | (note that z ₀ is negative under the assumption employed in the paper). Thus, the component of k_y = 0 dominantly contributes to the EEL spectrum. However, effects of non-zero k_y should be clarified in order to know the spatial coherence, which is also important for various applications as a novel radiation source of the SPR involving the PC’s.

Fig. 6. The dependence of the EEL spectrum on y ₀, the y-coordinate of the particle trajectory. y ₀ = 0 and 0.5a correspond to the trajectories lying on the mirror plane and thus the selection rule on the parity with respect to the planes holds. On the other hand, y ₀ = 0.25a corresponds to the trajectory lying between two adjacent mirror planes, where the selection rule is broken. The resonant frequencies at the intersection points between the (shifted) v-lines and the photonic dispersion curves of Fig. 5 are indicated by arrows. “e” stands for the y-even parity of the corresponding photonic band mode, whereas “o” stands for the y-odd parity.

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Suppose that we need a directional SPR normal to the PC slab (i.e. θ = ϕ = 90°). In this case the relevant eigenmodes are the quasi-guided modes at the Γ point. The modes at the Γ pointare classified according to the irreducible representation of C _4v, the point group of k at Γ. Among the five irreducible representations (A ₁,A ₂,B ₁,B ₂, and E) only the modes of the E representation, which are doubly degenerate, can couple to the external radiation [33]. Therefore, solely the E modes can mediate the directional SPR. We should note that this scenario of the directional SPR is broken if ωa/2πc > 1, where the Bragg diffraction channels open. In this case the angular distribution possesses additional peaks due to these channels.

By the numerical calculation of the photonic band structure, we found the quasi-guided modes of the E representation at ωa/2πc = 0.3101(odd), 0.3805(even), and 0.4127(odd), where the term “even” or “odd” refers to the parity with respect to the xy-plane bisecting the PC slab. Here, we focus on the mode at ωa/2πc = 0.4127, which is indicated by arrow in Fig. 5. Fig. 7 shows the intensity profile on the xz plane (y = 0) of the electric field induced by a moving charged particle with velocity v = 0.4127c along the trajectory (y ₀,z ₀) = (0, -0.5a). (ωa/2πc,k_ya/2π)= (0.4127,0) was taken such that the particle excites the above quasi-guided mode. At this frequency, the electric field is weakly bounded in the PC slab, and reveals the approximate mirror symmetries with respect to the plane z = 0. The obtained features of the spatial configuration indicate that the quasi-guided mode at the Γ point is certainly excited. We also show the frequency-resolved differential cross section (Eq.(27)) in Fig. 8, under the same parameters used above. The angular distribution is completely symmetric under the inversion ϕ → π - ϕ as a consequence of the symmetry of the radiation field. The directional SPR normal to the PC slab is certainly realized, taking account that θ = 90° is kineticly satisfied at this frequency. However, the angular distribution of the azimuthal angle is not sharp enough, mainly because the relevant quasi-guided modes around the Γ point have rather flat dispersion relation as was confirmed in Fig. 5. This implies that the modes near the Γ point in the y-direction are simultaneously excited. Therefore, in order to obtain a highly directional SPR, we need a fine tuning of frequency as well as an appropriate design of the PC slab in such a way that the relevant dispersion curve is not so flat.

Fig. 7. The intensity profile of the electric field on the xz plane (y = 0) induced by the moving charged particle with velocity v = 0.4127c along the trajectory (y ₀,z ₀) = (0,-0.5a). (ωa/2πc,k_ya/2π) = (0.4127,0) was assumed. The velocity was taken such that the particle excites the quasi-guide mode of the E representation at the Γ point. The interface between air and the dielectrics in the PC slab is represented by solid lines. The particle trajectory is indicated by arrow.

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4. Three-dimensional woodpile photonic crystal

Compared with 2d PC slab, 3d woodpile PC can control light more efficiently. PC slab controls light with the 2d periodicity in plane and the index guiding in the third direction. On the other hand, the woodpile is periodic also in the third direction, yielding the strong control of light in all spatial directions. In addition, the woodpile can have the omni-directional photonic band gap between the second and the third bands. The gap reaches as large as 20% of the gap-width/mid-gap ratio [34] and thus is robust against disorder. By introducing structural defects in the woodpile, a variety of defect modes appear. The quality factors of the defect modes increase exponentially with increasing number of the stacking layers surrounding the defects if there is no absorption loss [29]. These features are quite favorable for the SPR with high coherence. Thus, the characterization of the woodpile with the EEL and SPR spectra is in order.

First, let us consider the photonic band structure of a bulk woodpile PC. We assume the following parameters of the woodpile: w = 0.25a, d = a/(2√2), and ε = 12 + 0.01i (see Fig. 2). The thickness d was chosen in such a way that the woodpile has the exact face-centered cubic symmetry, when the system is infinite in all spatial directions. In actual numerical calculation of the EEL and SPR, we assume that the woodpile is infinite in plane, but has a finite height in the stacking direction. The photonic band structure of the woodpile is shown in Fig. 9, where the band structure along the z-direction (the stacking direction) is projected on the k_x axis provided k_ya/2π = 0 (a) and 0.25 (b). In Fig. 9 the shaded region corresponds to the bulk photonic eigenstates, while the blank region represents the (pseudo-)gaps. The wide band-gap around ωa/2πc = 0.4 is the omni-directional band gap of the woodpile PC. These projected band diagrams provide a basic information on the EEL and SPR spectra. In Fig. 9 we also show the dispersion curve of the planar defect mode. The planar defect is introduced by removing a mono-layer of the x-oriented stripe layer in the middle of the woodpile. The dispersion curve lies partially outside the light cone and thus the defect mode in this region is not detectable with a standard transmission measurement. However, the mode is detectable with the EEL spectroscopy even if it exists outside the light-cone. As shown in Fig. 9(a), the v-line of the evanescent wave accompanied by the moving charged particle with velocity v = 0.9c intersects the dispersion curve of the planar defect mode. Therefore, the defect mode at the intersection point is resonantly excited by the moving charged particle.

Fig. 8. The azimuthal angle distribution of the Smith-Purcell radiation (SPR) from the 2d PC slab at ωa/2πc = 0.4127. The same parameters as in Fig. 7 were assumed.

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Fig. 9. The photonic band structure of the 3d woodpile PC having the following parameters (See Fig. 2): w = 0.25a,d = a/(2√2), and ε = 12. The band structure along the the stacking direction is projected on the k_x -axis. Figs.(a) and (b) correspond to k_ya/2π = 0 and 0.25, respectively. The dispersion curve of the planar defect mode obtained by removing a mono-layer of x-oriented stripes from the bulk woodpile are shown by red lines. The boundary of the light-cone is represented by solid lines. The (shifted) v-lines of v = 0.9c in (a) and of v = 0.5c in (b) are also shown by dashed lines.

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Fig. 10. A cross-sectional view of the woodpile PC with the planar defect of removing a mono-layer of x-oriented stripes. The particle trajectory lies either inside the planar defect or outside the woodpile.

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Next, we focus on the EEL and SPR spectra in the woodpile PC with the planar defect. To be precise, a cross-sectional view of the system under study is given in Fig. 10. The planar defect is sandwiched by four stripe layers each in the upper and lower sides, and the top and bottom layers are composed of x-oriented stripes. The trajectory of the charged particle was taken to be either inside the planar defect or below the woodpile.

Fig. 11 shows the numerical result of the EEL and SPR spectra for the particle trajectory lying below the woodpile. The trajectory of the charged particle was taken to be (y ₀,z ₀) = (-0.25a, -d), where y ₀ and z ₀ are measured from the center of a x-oriented stripe of the bottom layer. Here, we assumed k_y = 0 and v = 0.9c to compare with Fig. 9(a). As shown in Fig. 11, the EEL spectrum has so many peaks particularly at high frequencies above the photonic band gap. On the other hand, below the band gap, the peaks are well separated in frequency. These peaks arise when the (shifted) v-lines lie in the shaded region of the band diagram, and thus the peaks indicate the excitation of the bulk eigenstates of the woodpile PC. This feature on the peak distribution in frequency is quite reasonable taking account that several bands overlap at high frequencies, while there are only two bands below the band gap. In the band gap we found a sharp peak at ωa/2πc = 0.3815, whose quality factor is about 1700. The frequency of this peak shifts gradually as we change the velocity of the particle. By plotting the peak frequency as a function of the wave number k_x = ω/v, we obtained the dispersion curve denoted by red lines in Fig. 9. This curve coincides with the dispersion curve of the planar defect mode, which can be obtained from the resonance in the transmittance through the woodpile PC by using the S-matrix formalism. Also, the curve coincides very well with that obtained with the the plane-wave expansion method using a supercell approach. Therefore, the peak is due to the planar defect mode localized in the defect layer. On the other hand, the SPR spectrum has the threshold frequency, below which the SPR is kineticly forbidden because of the absence of the open diffraction channels. Above the threshold, the SPR spectrum is almost same as the EEL spectrum.

Fig. 11. The EEL and SPR spectra at k_y = 0 in the woodpile with the planar defect. The charged particle moves with velocity v = 0.9c along the trajectory given by (y ₀,z ₀) = (-0.25a,- d) (see Fig. 10). The omni-directional photonic band gap (PBG) is indicated by arrow.

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Fig. 12 shows the intensity profile of the electric field at ωa/2πc = 0.3815 (denoted by P ₁ in Fig. 9(a)) induced by the moving charged particle. This Fig. indicates clearly that the planar defect mode is resonantly excited by the particle moving outside the woodpile PC.

Next, we turn our attention to a directional SPR using the woodpile PC. The planar defect mode obtained above can not couple with propagating external radiation, because it exists outside the light-cone. Instead, if the defect mode concerned lies inside the light-cone, having sufficiently high quality factor and large coupling to the evanescent radiation accompanied by the moving charged particle, SPR of high coherence can be generated. Let us consider the case of k_ya/2π = 0.25. The photonic band structure at k_ya/2π = 0.25 is shown in Fig. 9 (b). The shifted v-line of v = 0.5c intersects the dispersion curve of the planar defect mode at ωa/2πc = 0.4030 inside the light-cone.

We show in Fig. 13 the angular distributions of the SPR at this frequency for two different trajectories of the moving charged particle. One is inside the defect layer (a), and the other is outside the woodpile PC (b). The coordinate of the particle trajectory was taken to be (y ₀,z ₀) = (a/2,d/4) and (y ₀,z ₀) = (0, -d) in (a) and (b), respectively (see Fig. 10). In the former case the charged particle couples to the defect mode very strongly, because the overlap between the defect mode and the evanescent radiation of the charged particle is very large. On the other hand, in the latter case, the coupling is weak because the defect layer is well separated from the particle trajectory. Even if a fine tuning of frequency is made, the absorption loss due to the finite imaginary part in e reduces the coupling. In Fig. 13(a) the SPR has very high directivity toward the angles (θ,ϕ) ≈ (119°, ±45°), and (119°,±135°) determined by Eq.(26) with (ωa/2πc,k_ya/2π) = (0.403,0.25). Moreover, it has much higher SPR intensity at this solid angle compared to the latter case. In Fig. 13(b) most power of the SPR is reflected with wide angle, though the transmitted SPR has high directivity. The intensity of the transmitted SPR is very small and thus it is difficult to detect the directional SPR. Therefore, we can say that the particle moving inside the defect can induce a SPR of very high coherence by using the defect mode.

Finally, we should note that the directional SPR is quite difficult to realize if bulk eigenstates of the woodpile PC are used. The bulk eigenstates have 3d dispersion relations ω = ω_n (k_x ,k_y ,k_z ). As was shown in this paper, the SPR is strongly enhanced when the shifted v-lines intersect the dispersion curves. The matching condition between the dispersion relations and the shifted v-lines generally yields a one-dimensional solution in the momentum space at a given frequency. Thus, the SPR becomes “conical”, like as in the Cherenkov radiation. On the other hand, the planar defect mode has the 2d dispersion relation ω = ω_n (k_x ,k_y ), in which k_z disappears owing to the broken translational invariance and to localization effects. In this case the matching condition yields a zero-dimensional (point-like) solution. Therefore, the SPR becomes directional at a given frequency.

Fig. 12. The intensity profile of the electric field at ωa/2πc = 0.3815, where the v-line of v = 0.9c intersects the dispersion curve of the planar defect mode in Fig. 9 (a) (the intersection point is indicated by P ₁). The electric field is induced by the moving charged particle indicated by arrow outside the woodpile. Red lines stand for the boundary of the stripes. (y ₀,z ₀) = (0,-d) was assumed.

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5. Summary

To summarize, we have presented a theoretical analysis of the EEL and SPR spectra for a charged particle moving near 2d or 3d PC’s. A detailed analysis was made for a 2d square PC slab of cylindrical air holes fabricated in a thin dielectric slab and for a 3d woodpile PC of rectangular stripes, by employing the S-matrix method. We have clarified that the EEL and SPR spectra are deeply correlated with the photonic band structure in these PC’s with and without a structural defect. By combining the selection rule depending on the trajectory of the charged particle, the spectra can be used as a probe of the photonic band structure. Moreover, we have shown that a directional SPR can be realized by using the quasi-guided modes in the PC slab and the planar defect mode of the woodpile PC.

Fig. 13. Frequency-resolved angular distribution at ωa/2πc = 0.403 of the Smith-Purcell radiation from the woodpile PC with the planar defect (See Fig. 10). The corresponding defect mode is indicated as P ₂ in Fig. 9(b). The particle moves with velocity v = 0.5c inside the planar defect (a) or outside the woodpile (b).

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Acknowledgments

This work was supported by “Promotion of Science and Technology” from the Ministry of Education, Sports, Culture, Science, and Technology of Japan.

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Electron energy loss and Smith-Purcell radiation in two- and three-dimensional photonic crystals

Abstract

1. Introduction

2. Formalism

3. Two-dimensional photonic crystal slab

4. Three-dimensional woodpile photonic crystal

5. Summary

Acknowledgments

References and links

Cited By

Figures (13)

Equations (28)

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