Abstract

A theoretical description of the electron energy loss and the Smith-Purcell radiation is presented for an electron moving near a two-dimensional photonic crystal slab and a three-dimensional woodpile photonic crystal. The electron energy loss and the Smith-Purcell radiation spectra are well correlated with the photonic band structures of these crystals and thus can be used as a probe of them. In particular, there is a selection rule concerning the symmetries of the photonic band modes to be excited when the electron moves in a mirror plane of the crystals. In the woodpile, a highly directional Smith-Purcell radiation is realized by using the planar defect mode inside the complete band gap.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. K. Yamamoto, R. Sakakibara, S. Yano, Y. Segawa, Y. Shibata, K. Ishi, T. Ohsaka, T. Hara, Y. Kondo, H. Miyazaki, F. Hinode, T. Matsuyama, S. Yamaguti, and K. Ohtaka, �??Observation of millimeter-wave radiation generated by the interaction between an electron beam and a photonic crystal,�?? Phys. Rev. E 69, 045601 (2004).
    [CrossRef]
  25. S. P. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, �??Control of light emission by 3D photonic crystals,�?? Science 305, 227�??229 (2004).
    [CrossRef] [PubMed]
  26. M. Okano and S. Noda, �??Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,�?? Phys. Rev. B 70, 125105 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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  29. T. Ueta, K. Ohtaka, N. Kawai, and K. Sakoda, �??Limits on quality factors of localized defect modes in photonic crystals due to dielectric loss,�?? J. Appl. Phys. 84, 6299�??6304 (1998).
    [CrossRef]
  30. T. Ochiai and K. Sakoda, �??Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,�?? Phys. Rev. B 63, 125107 (2001).
    [CrossRef]
  31. T. Ochiai and K. Sakoda, �??Nearly free-photon approximation for two-dimensional photonic crystal slabs,�?? Phys. Rev. B 64, 045108 (2001).
    [CrossRef]
  32. K. Ohtaka, J. Inoue, and S. Yamaguti, �??Derivation of the density of states of leaky photonic bands,�?? Phys. Rev. B 70, 035,109 (2004).
    [CrossRef]
  33. P. Paddon and J. F. Young, �??Two-dimensional vector-coupled-mode theory for textured planar waveguides,�?? Phys. Rev. B 61, 2090�??2101 (2000).
    [CrossRef]
  34. B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, �??Theoretical study of photonic band gaps in woodpile crystals,�?? Phys. Rev. E 67, 066601 (2003).
    [CrossRef]

J. Appl. Phys. (1)

T. Ueta, K. Ohtaka, N. Kawai, and K. Sakoda, �??Limits on quality factors of localized defect modes in photonic crystals due to dielectric loss,�?? J. Appl. Phys. 84, 6299�??6304 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Nature (2)

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R.Wendt, G. A. Vawter,W. Zubrzycki, H. Hou, and A. Alleman, �??Three-dimensional control of light in a two-dimensional photonic crystal slab,�?? Nature 407, 983�??986 (2000).
[CrossRef] [PubMed]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, �??A three-dimensional photonic crystal operating at infrared wavelengths,�?? Nature 394, 251�??253 (1998).
[CrossRef]

Opt. Spectrosc. (1)

K. Ohtaka and S. Yamaguti, �??Theoretical study of the Smith-Purcell effect involving photonic crystals,�?? Opt. Spectrosc. 91, 477�??483 (2001).
[CrossRef]

Phys. B (1)

F. J. García de Abajo and L. A. Blanco, �??Electron energy loss and induced photon emission in photonic crystals,�?? Phys. Rev. B 67, 125108 (2003).
[CrossRef]

Phys. Rev. (1)

S. J. Smith and E. M. Purcell, �??Visible light from localized surface charges moving across a grating,�?? Phys. Rev. 92, 1069 (1953).
[CrossRef]

Phys. Rev. A (1)

D. E. Wortman, R. P. Leavitt, H. Dropkin, and C. A. Morrison, �??Generation of millimeter-wave radiation by means of a Smith-Purcell free-electron laser,�?? Phys. Rev. A 24, 1150�??1153 (1981).
[CrossRef]

Phys. Rev. B (12)

F. J. García de Abajo, A. Rivacoba, N. Zabala, and P. M. Echenique, �??Electron energy loss spectroscopy as a probe of two-dimensional photonic crystals,�?? Phys. Rev. B 68, 205105 (2003).
[CrossRef]

T. Ochiai and K. Ohtaka, �??Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals. I. Formalism and surface plasmon polariton,�?? Phys. Rev. B 69, 125106 (2004).
[CrossRef]

T. Ochiai and K. Ohtaka, �??Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals. II. Photonic band effects,�?? Phys. Rev. B 69, 125107 (2004).
[CrossRef]

D. M. Whittaker and I. S. Culshaw, �??Scattering-matrix treatment of patterned multilayer photonic structures,�?? Phys. Rev. B 60, 2610�??2618 (1999).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, �??Quasiguided modes and optical properties of photonic crystal slabs,�?? Phys. Rev. B 66, 045102 (2002).
[CrossRef]

S. Yamaguti, J. Inoue, O. Haeberlé, and K. Ohtaka, �??Photonic crystals versus diffraction gratings in Smith-Purcell radiation,�?? Phys. Rev. B 66, 195202 (2002).
[CrossRef]

T. Ochiai and K. Sakoda, �??Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,�?? Phys. Rev. B 63, 125107 (2001).
[CrossRef]

T. Ochiai and K. Sakoda, �??Nearly free-photon approximation for two-dimensional photonic crystal slabs,�?? Phys. Rev. B 64, 045108 (2001).
[CrossRef]

K. Ohtaka, J. Inoue, and S. Yamaguti, �??Derivation of the density of states of leaky photonic bands,�?? Phys. Rev. B 70, 035,109 (2004).
[CrossRef]

P. Paddon and J. F. Young, �??Two-dimensional vector-coupled-mode theory for textured planar waveguides,�?? Phys. Rev. B 61, 2090�??2101 (2000).
[CrossRef]

M. Okano and S. Noda, �??Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,�?? Phys. Rev. B 70, 125105 (2004).
[CrossRef]

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

Phys. Rev. E (7)

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, �??Theoretical study of photonic band gaps in woodpile crystals,�?? Phys. Rev. E 67, 066601 (2003).
[CrossRef]

K. Yamamoto, R. Sakakibara, S. Yano, Y. Segawa, Y. Shibata, K. Ishi, T. Ohsaka, T. Hara, Y. Kondo, H. Miyazaki, F. Hinode, T. Matsuyama, S. Yamaguti, and K. Ohtaka, �??Observation of millimeter-wave radiation generated by the interaction between an electron beam and a photonic crystal,�?? Phys. Rev. E 69, 045601 (2004).
[CrossRef]

O. Haeberlé, P. Rullhusen, J. M. Salomé, and N. Maene, �??Calculations of Smith-Purcell radiation generated by electrons of 1-100 Mev,�?? Phys. Rev. E 49, 3340�??3352 (1994).
[CrossRef]

K. Ishi, Y. Shibata, T. Takahashi, S. Hasebe, M. Ikezawa, K. Takami, T. Matsuyama, K. Kobayashi, and Y. Fujita, �??Observation of coherent Smith-Purcell radiation from short-bunched electrons,�?? Phys. Rev. E 51, R5212�??R5215 (1995).
[CrossRef]

Y. Shibata, S. Hasebe, K. Ishi, S. Ono, M. Ikezawa, T. Nakazato, M. Oyamada, S. Urasawa, T. Takahashi, T. Matsuyama, K. Kobayashi, and Y. Fujita, �??Coherent Smith-Purcell radiation in the millimeter-wave region from a short-bunch beam of relativistic electrons,�?? Phys. Rev. E 57, 1061�??1074 (1998).
[CrossRef]

F. J. García de Abajo, �??Smith-Purcell radiation emission in aligned nanoparticles,�?? Phys. Rev. E 61, 5743�??5752 (2000).
[CrossRef]

Z. Y. Li and L. L. Lin, �??Photonic band structures solved by a plane-wave-based transfer-matrix method,�?? Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

F. J. García 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]

G. Doucas, J. H. Mulvey, M. Omori, J. Walsh, and M. F. Kimmitt, �??First observation of Smith-Purcell radiation from relativistic electrons,�?? Phys. Rev. Lett. 69, 1761�??1764 (1992).
[CrossRef] [PubMed]

Science (2)

C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, Science 299, 368�??371 (2003).
[CrossRef] [PubMed]

S. P. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, �??Control of light emission by 3D photonic crystals,�?? Science 305, 227�??229 (2004).
[CrossRef] [PubMed]

Other (1)

J. Pendry, Low Energy Electron Diffraction (Academic, London, 1974).

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Figures (13)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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.

Fig. 4.
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.

Fig. 5.
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. ky = 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.

Fig. 6.
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.

Fig. 7.
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,kya/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.

Fig. 8.
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.

Fig. 9.
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 kx -axis. Figs.(a) and (b) correspond to kya/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.

Fig. 10.
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.

Fig. 11.
Fig. 11.

The EEL and SPR spectra at ky = 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.

Fig. 12.
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.

Fig. 13.
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).

Equations (28)

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E ( x ; ω ) = dk y 2 π ε ± ( ω v , k y ) e i K ± ( x x 0 ) ,
ε ± ( ω v , k y ) = μ 0 2 1 Γ ( 1 1 q 2 ( ω v ) 2 ) 1 Γ q 2 ω v k y 1 q 2 ω v ,
K ± = ( ω v , k y , ±Γ ) , Γ = q 2 ( ω v ) 2 k y 2 q = ω c ,
E tra ( x ; ω ) = G π a π a dk y 2 π e i K G + x t G ± ( ω , k y ) ,
t G + ( ω , k y ) = G S + + ( G , G ) δ G ' x , G x 0 ε G + e i k G ' + x 0 ,
ε G ± = ε ± ( k x + G x , k y + G y ) ,
K G ± = ( k x + G x , k y + G y ± Γ G ) ,
Γ G = q 2 ( k x + G x ) 2 ( k y + G y ) 2 .
E ref ( x ; ω ) = G π a π a dk y 2 π e i K G x t G ( ω , k y ) ,
t G ( ω , k y ) = G S + ( G , G ) δ G x , G x 0 ε G + e i K G + x 0 .
W el = 2 π 0 P pel ( ω ) ,
P el ( ω ) = e 2 L Re [ dte iωt v E ref ( x t ; ω ) ] ,
= π a π a dk y 2 π P el ( ω , k y ) ,
P el ( ω , k y ) = e 2 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 ]
W sp = 2 π 0 P sp ( ω ) ,
P sp ( ω ) = 1 a 0 a dx dy 1 2 Re ( E * ( x ; ω ) × H ( x ; ω ) ) z
= π a π a dk y 2 π P sp ( ω , k y ) ,
P sp ( ω , k y ) = 1 2 μ 0 ω G open Γ G ( t G + 2 + t G 2 ) ,
E ind ( x ; ω ) = G π a π a 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 ,
E tra , ± ( x ; ω ) = G π a π a dk y 2 π e i K G ± x t G ± ( ω , k y ) ,
t G + = [ S ++ u ( ε + + a + ) ] G ,
t G = [ S l ( ε + a ) ] G .
P el ( ω , k y ) = e 2 G y Re ( a G x 0 , G y + + a G x 0 , G y ) x .
K G ± = q ( cos θ , sin θ cos ϕ , sin θ sin ϕ ) .
2 W sp ω ϕ ( sin θ cos ϕ ) 2 t G ± ( ω , k y ) 2 ,
P el ( ω , k y ) = e 2 Re [ e 2 i Γ z 0 ( + ) x ] ,

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