## Theory of unconventional Smith-Purcell radiation in finite-size photonic crystals

Optics Express, Vol. 14, Issue 16, pp. 7378-7397 (2006)

http://dx.doi.org/10.1364/OE.14.007378

Acrobat PDF (3735 KB)

### Abstract

Unusual emission of light, called the unconventional Smith-Purcell radiation (uSPR) in this paper, was demonstrated from an electron traveling near a finite photonic crystal (PhC) at an ultra-relativistic velocity. This phenomenon is not related to the accepted mechanism of the conventional SPR and arises because the evanescent light from the electron has such a small decay constant in the ultra-relativistic regime that it works practically as a plane-wave probe entering the PhC from one end. We analyze the dependence of the SPR spectrum on the velocity of electron and on the parity of excited photonic bands and show, for PhCs made up of a finite number of cylinders, that uSPR probes the photonic band structure very faithfully.

© 2006 Optical Society of America

## 1. Introduction

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

3. J. M. Wachtel, “Free-electron lasers using the Smith-Purcell effect,” J. Appl. Phys. **50**, 49–56 (1979). [CrossRef]

4. 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]

5. A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation-pattern of Smith-Purcell radiation,” J. Opt. Soc. Am. B **1**, 723–728 (1984). [CrossRef]

6. I. Shih, W. W. Salisbury, D. L. Masters, and D. B. Chang, “Measurements of Smith-Purcell radiation,” J. Opt. Soc. Am. B **7**, 345–350 (1990). [CrossRef]

7. 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]

8. 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]

9. P. M. van den Berg, “Smith-Purcell radiation from a point charge moving parallel to a reflection grating,” J. Opt. Soc. Am. **63**, 1588–1597 (1973). [CrossRef]

10. 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]

11. Y. Shibata, S. Hasebe, K. Ishi, S. Ono, M. Ikezawa, T. Nakazato, M. Oyamada, S. Urasawa, T. Takahashi, T. Mat-suyama, 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]

12. J. H. Brownell, J. Walsh, and G. Doucas, “Spontaneous Smith-Purcell radiation described through induced surface currents,” Phys. Rev. E **57**, 1075–1080 (1998). [CrossRef]

13. R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. **48**, 928–936 (1935). [CrossRef]

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

15. F. J. Garcìa de Abajo, “Smith-Purcell radiation emission in aligned nanoparticles,” Phys. Rev. E **61**, 5743–5752 (2000). [CrossRef]

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

17. 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]

18. F. J. Garcìa de Abajo and L. A. Blanco, “Electron energy loss and induced photon emission in photonic crystals,” Phys. Rev. B **67**, 125108 (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. 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]

21. 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(R) (2004). [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]

## 2. Kinetics of conventional and unconventional SPRs

*a*) in the x direction with cylinder axes in the

*z*direction. An electron travels near the PhC in a trajectory parallel to the

*x*axis with velocity

*v*and impact parameter

*b*. We obtain the SPR spectrum from this system as a sum of the plane-wave signals generated by the scattering of the evanescent light emitted by the electron. The whole process of multiple scattering among a finite number of cylinders is dealt with compactly by the multiple-scattering theory of radiation using the vector cylindrical waves as a basis of representation [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]

*x*direction. A traveling electron accompanies the radiation field that is a superposition of evanescent waves with respect to frequency

*ω*and wave number

*k*

_{z}in the

*z*direction [16

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

*v*≤

*c*. The imaginary part |Γ| determines the spatial decay of the evanescent wave incident on the PhC. In what follows, it is important to remember the feature of

*k*

_{z}= 0 that, in the ultimate limit

*v*→

*c*, |Γ| tends to zero. Since

*ω*and

*k*

_{z}are conserved quantities in the geometry of Fig. 1, the evanescent waves with different

*ω*and

*k*

_{z}are independent in the whole scattering process. Thus, the incident light of ft) and

*k*

_{z}leaves the PhC, after being scattered, with the same

*ω*and

*k*

_{z}. Therefore, the SPR signals observed in the

*xy*plane may be analyzed by setting

*k*

_{z}= 0 everywhere. Since we are now dealing with a perfect periodicity extending from - ∞ to ∞, we obtain the SPR signal in the form of Bragg-scattered waves summed over the diffraction channels. The channels are specified by the wave vector

*h*= 2

*πn*/

*a*(

*n*: integer) is a reciprocal lattice point of the PhC in the

*x*direction. Before the scattering by the PhC,

*ω*and

*k*

_{x}, the

*x*component of the wavevector of light, satisfy the relation

*k*

_{x}=

*ω*/

*v*. The line

*ω*=

*vk*

_{x}, called the

*v*line in this paper, lies outside the light cone in the phase space (

*k*

_{x},

*ω*). After the scattering, the

*x*component of the light of channel

*h*becomes

*k*

_{x}=

*ω*/

*v*-

*h*. The shifted v line defined by this equation is inside the light cone in a certain frequency range. In that frequency range we can detect the SPR signal in this channel at a far-field observation point. The propagating direction of the SPR signal of

*ω*is given by

*k*

_{x}

*,k*

_{z}

*, ω*) space. The peak position determines the dispersion relations

*ω*=

*ω*

_{n}(

*k*

_{x}

*,k*

_{z}) of the quasi-guided PhB modes. Imagine temporarily

*k*

_{z}= 0, for brevity. The presence of the modes in the (

*k*

_{x}

*, ω*) space significantly affects the SPR spectrum by causing a sharp resonance when the dispersion curves of PhBs intersect the shifted v lines. The resonance becomes sharper as the quality factor of the relevant PhB modes increases [16

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

17. 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]

*k*

_{x}, with an Umklapp allowance taken into account. In an actual PhC with a finite number

*N*of cylinders, the periodicity or the translational invariance of the whole system is lost at the sample edges. One way to take account of the finiteness of

*N*is to treat

*k*

_{x}as defined only approximately with a width of the order of Δ

*k*

_{x}≃ 2

*π*/(

*Na*) [2]. In this approach, the shifted

*v*lines are considered to have the finite width and the PhB dispersion relation will become detectable within this allowance centering on the shifted v lines of open channels. In reality, however, the uSPR signals appear in the phase space (

*k*

_{x},

*ω*) with a much larger distribution than this straightforward 1/

*N*blurring [22].

*v*lines. In addition, since

*v*≈

*c*, the shifted

*v*lines coincide with the threshold lines for the opening of a new Bragg diffraction channel. The channel opening often leaves a singular trace due to Wood’s anomaly in the line shape of wave scattering. Thus, on our shifted

*v*lines, Wood’s anomaly will occur, together with the resonance peaks of the cSPR associated with the PhB excitation. In this way, the spectra of the c and u SPRs reveal a quite rich structure when the electron is ultra-relativistic.

*x*direction, that is, the direction of the periodicity of the PhC under consideration. If the velocity vector v of the electron is given by

*k*

_{z}depends on frequency and is given by

*k*

_{z}=

*v*

_{z}

*ω*/

*v*

^{2}. Within the conventional theory the SPR acquires a significant enhancement when the following three conditions are fulfilled:

*v*line at nonzero

*v*

_{z}. As in the case of

*v*

_{z}= 0 (

*α*= 0°), this scenario of the SPR is insufficient for an ultra-relativistic electron. In this case the evanescent wave accompanied by the electron can be effectively treated as a plane wave propagating parallel to v. This highlights the role of the sample edge of the finite-size PhC, namely, the broken translational invariance, and the photonic band modes on the entire plane of

*k*

_{z}= (

*v*

_{z}/

*v*

^{2})

*ω*are excited, not necessary restricted on the shifted

*v*lines. In the limiting case of vanishing

*v*

_{x}(

*α*= 90°), the electron travels parallel to the cylindrical axis. There, no propagating radiation is generated from the PhC, as far as the cylinders have infinite length in the

*z*direction. This is due to the perfect translational invariance along the axis. In actual PhC, however, this translational invariance is broken, yielding a sort of the diffraction radiation. Thus, as

*α*varies from 0° to 90°, the conventional theory of the SPR, which assumes the translational invariance both in the

*x*and

*z*direction, predicts a gradual disappearance of the SPR. On the other hand, the uSPR gives a novel radiation irrespective of

*α*, in which the broken translational invariance in the

*x*direction is highlighted at small

*α*, and that in the

*z*direction is highlighted around

*α*= 90°.

*N*≥ 8 will be enough according to our experience. In contrast to the value of

*N*in the

*x*direction, however, we are considering a system having small size in the

*y*direction, such as a PhC made of a monolayer or stacked layers of several monolayers. The finite size in the

*y*direction needs to be considered explicitly to obtain the band structure of our PhCs.

## 3. Typical example of conventional and unconventional SPRs

*ε*= 2.05). Foraradius-to-periodicity ratio

*r/a*= 0.5, Fig. 3 depicts the band structure of the monolayer of an infinite number of cylinders. The band structure inside the light cone was obtained by plotting the peak frequencies of the ODOS, which were calculated as a function of

*k*

_{x}and

*k*

_{z}[28

28. K. Ohtaka, J. Inoue, and S. Yamaguti, “Derivation of the density of states of leaky photonic bands,” Phys. Rev. B **70**, 035109 (2004). [CrossRef]

*ω*plane. In Fig. 3,

*k*

_{z}= 0 is assumed, so that the PhB modes are decomposed into purely transverse-electric (TE) and transverse-magnetic (TM) modes. Only the band structure of the TE modes is presented, because the incident evanescent wave is TE-polarized at

*k*

_{z}= 0. The PhB modes are further classified by parity with respect to the mirror plane

*y*= 0. In Fig. 3 the even (odd) parity modes are indicated by red (green) circles. We should note that PhB modes having their dispersion curves disconnected in Fig. 3 are the ones obtained from the ODOS peaks, which are often too broad to identify the peak position.

*x*direction. We used the parameters

*v*= 0.99999

*c*,

*ϕ*= 180° and

*b*= 3.33

*a*and assumed that the radiation was observed in the

*xy*plane (

*k*

_{z}= 0), as actually encountered in the millimeter-wave SPR experiments carried out recently [21

21. 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(R) (2004). [CrossRef]

*v*lines, which are almost parallel to the light line

*ω*=

*ck*

_{x}. Figure 4 presents the reflected cSPR spectra along the shifted

*v*line of

*h*= 1 and 2 (in units of 2π/

*a*). The peaks of the cSPR spectrum arise at the frequency where the shifted

*v*lines

*k*

_{x}=

*ω/v - h*intersect the PhB structure given in Fig. 3. Several arrows are drawn at the peak positions in Fig. 4 and, to identify each of the peaks, horizontal arrows are added in Fig. 3 at the corresponding positions in phase space. Comparing these two figures, we see that the peak lowest in frequency arises from the excitation of an odd-parity PhB mode. Thus, the even selection-rule for the parity of the excited PhB modes does not hold for cSPR, though it somewhat affects their spectral shapes.

*N*considered explicitly, is obtained over the entire (

*k*

_{x},

*ω*) space by summing all the amplitudes of the multiply scattered light from the

*N*cylinders [19

**69**, 125106 (2004). [CrossRef]

*N*= 21 is given in Fig. 5, for the same parameters of

*r/a*and e as used in Fig. 4. The angle

*θ*- and frequency

*ω*-resolved reflected SPR intensity is mapped onto the (

*k*

_{x}

*, ω*) plane through the relation

*k*

_{x}= (

*ω/c*) cos

*θ*. To be precise, |

*f*

^{M}(

*θ*)|

^{2}with -

*π*≤

*θ*≤ 0 defined in Eq. (33) of Ref [19

**69**, 125106 (2004). [CrossRef]

- the shifted
*v*lines, - the curves whose slopes are positive and less than 1,
- the curves whose slopes are negative,
- the forward light-line (
*ω*=*ck*_{x}), and - (E) the flat lines terminated on the backward light-line
*ω*= -*ck*_{x}).

*N*= ∞ but bounded at one end, because what matters in the above discussion is the presence of the left edge of PhC as an entrance surface of a wave propagating in the

*x*direction.

*v*= 0.5

*c*, which is a typical value for the electron velocity used in scanning electron microscopes. The parameters except

*v*and

*b*are the same. As above, we compare two spectra of

*N*= ∞ and

*N*= 21.

*N*= ∞ is given in Fig. 7. The spectra reveal a marked resonance at

*ωa*/2

*πc*= 0.621. The line shape of the resonance is asymmetric as a function of frequency. As indicated by arrows, each agreeing precisely with those given to the shifted

*v*line of Fig. 3, the cSPR peaks all appear exactly at the intersection points of the shifted

*v*line of

*v*= 0.5

*c*with the PhB dispersion curves.

*N*= 21 is given in Fig. 8, with the superposition of the PhB structure (of

*N*= ∞). We see at once that high intensity SPR appears only on the shifted

*v*lines, although very weak structures reminiscent of the finiteness of our PhC are still seen off the shifted

*v*line. This is in clear contrast to the ultra-relativistic spectra, where marked signals of uSPR existed definitely off the shifted

*v*lines. The signals on the shifted

*v*line of

*h*= 1 have a resonance peak at (

*k*

_{x}

*a*/2

*π*,

*ωa*/2

*πc*) = (0.24,0.62). This frequency is almost identical to that of the resonance obtained for

*N*= ∞ shown in Fig. 7. Also, we can perceive the asymmetry of the line shape along the shifted

*v*line, as in the cSPR spectrum of

*N*= ∞. Therefore, we may conclude that, for slower velocities such as

*v*= 0.5

*c*, the SPR of the finite PhC can be understood sufficiently well using the theory of cSPR, based on the assumption

*N*= ∞. The uSPR signals are suppressed as follows. The light of

*v*= 0.5

*c*is literally evanescent with an appreciable decay constant |Γ|, so that, while passing through the PhC in the +

*y*direction, the incident light decays much and sees only the surface region of cylinders. Accordingly, the picture of a plane wave with wavevector in the

*x*direction no longer holds and the conventional theory of SPR covers all the features.

## 4. Properties of the unconventional SPR

*N*) is crucial in the uSPR. Taking account that the uSPR must vanish in the system of the perfect translational invariance, it is interesting to investigate the

*N*-dependence of the uSPR in detail. The number of stacking layers (

*N*

_{l}) is also an important factor because ODOS and thus the PhB structure depends crucially on

*N*

_{l}. Dielectric constant

*ε*and radius

*r*of the cylinders are other factors that significantly influence the PhB structure. However, the effects of changing

*r*are covered, to some extent, by those of

*ε*. The impact parameter

*b*is not essential, as seen in the following expression for the total emission power

*W*of SPR, whose

*b*dependence is collected into a simple scaling law [19

**69**, 125106 (2004). [CrossRef]

*P*

_{em}(

*ω,k*

_{z})|

_{b}is the

*ω*- and

*k*

_{z}-resolved emission power for an impact parameter b and

*b*

_{0}is a reference impact parameter chosen arbitrarily. Therefore, uSPR and cSPR change in a straightforward way as

*b*varies, with the underlying physics unaltered. In the following subsections, therefore, five parameters,

*v*,

*ε,N,N*

_{l}, and

*ϕ*, are varied in this order to see how each affects the spectrum.

### 4.1. Velocity

*v*= 0.5

*c*is understood using the theory of cSPR, while at

*v*= 0.99999

*c*the uSPR also plays an important role. We shall examine how the conventional picture fails with varying electron velocity. An obvious but nonessential

*v*-dependence is an increase of the SPR intensity due to the

*v*dependence of the decay-constant |Γ|; if impact parameter

*b*is fixed, the overall SPR spectrum behaves as exp(-2|Γ|

*b*). To eliminate this trivial v-dependence, we have set

*βγ*) for

*k*

_{z}= 0.

*N*= 21 are shown in Figs. 9 (a) and (b) for

*v*= 0.7

*c*(

*γ*= 1.4) and in (c) and (d) for 0.99

*c*(

*γ*=7.09), along with the PhB structure. Panels (a) and (c) show only the SPR intensity, while they are superposed by the PhB structure in panels (b) and (d).

*v*= 0.7

*c*, there is a marked bright line along the shifted

*v*line of

*h*= 1. Along the line, the intensity contrast of the SPR is quite strong at low frequencies. In particular a point-like resonance is seen at

*ωa*/2

*πc*≃ 0.745. As panel (b) shows, this resonance arises just at a crossing between the dispersion curve of an even-parity PhB and the shifted

*v*line of

*h*= 1. Therefore, this is a type (A) signal according to the classification of the last section.

*k*

_{x}approaches the backward light line

*ω*= -

*ck*

_{x}. This feature is common to all the horizontal streaks appearing at the frequencies of the pseudo gaps. These are signals of type (E) of the uSPR. We should note that the PhB mode, which crosses the shifted

*v*lines, has a negative group velocity, and the excited PhB mode propagates in the -

*x*direction. A backward-oriented diffraction taking place at the left edge of the PhC explains the tendency towards the line

*ω*= -

*ck*

_{x}. Analogous flat lines exist, for instance, at

*ωa*/2

*πc*≃ 1.09.

*v*lines. The curves are in fact coincident with the dispersion curves of quasi-guided PhB of the even parity. Therefore, they are type (B) signals. Note that the odd-parity PhB dispersion curves are also visible, with reduced strength as compared to the even-parity PhBs. Altogether, at

*v*= 0.7

*c*, cSPR coexists with uSPR and odd-parity PhBs are seen in the uSPR spectrum, with weaker intensity than even-parity PhBs, however. Combining this result with what we have seen in Sec. III for

*v*= 0.5

*c*and

*v*= 0.99999

*c*, we may conclude that, as

*v*increases from

*v*= 0.5

*c*, the uSPR becomes visible and the even-parity selection rule of uSPR is less stringent at non-ultra-relativistic velocities.

*v*= 0.99

*c*indeed confirms this conclusion. At

*v*= 0.99

*c*, several bright curves arise in Figs. 9 (c) and (d) with little intensity contrast along the PhB dispersion curves. This is the type (B) signal. We can observe odd-parity excitation of weak intensity. Therefore, although the even-parity selection rule is indeed dominant, it is somewhat relaxed for

*v*= 0.99

*c*. On the shifted

*v*line, there are signals of cSPR, as theory predicted for type (A) features in Sec. III.

*v*, |Γ| increases to make the incident evanescent light decay more quickly when passing the monolayer. This increases the asymmetry of the evanescent wave with respect to the mirror plane and makes the even-parity selection rule less effective. The degree of the symmetry of the input wave may be given by the factor exp(-|ω|2

*r*), called here the symmetry factor, which measures the decay of the evanescent wave while traversing the PhC in the +

*y*direction. If this factor is unity, the evanescent light seen by the PhC is mirror-symmetric. At

*v*= 0.99

*c*, the symmetry factor is 0.408 at

*ωa*/2

*πc*= 1 and too small to guarantee strictly the even-parity selection rule. Therefore, odd-parity PhBs are allowed somewhat as uSPR signals.

*v*are briefly summarized without giving numerical results. At

*v*= 0.9

*c*, when the symmetry factor is 0.047, cSPR and uSPR coexist and odd-parity PhBs are seen in the latter. At

*v*= 0.999

*c*, the asymmetry factor increases to 0.755. The intensity map gradually tends to the case of

*v*= 0.99999

*c*with the symmetry factor 0.972; signals along the odd-parity PhBs disappear, leaving behind only the even-parity signals as type (B) signals. The horizontal bright streaks appear solely in the regions of the pseudo-gaps of even-parity bands.

*v*

_{z}. The critical velocity of the electron, above which the uSPR begins to emerge does not change so much by non-zero

*v*

_{z}. An important point is that at ultra-relativistic velocities the evanescent wave can be effectively regarded as a plane wave. This is not controlled by

*v*

_{z}, but is controlled by

*v*, the magnitude of the velocity vector. However, other features of the uSPR changes as discussed in the previous section.

*ωa*/2

*πc*~ 1, the uSPR is conspicuous when v exceeds 0.7c, and the even-parity selection rule holds progressively better as

*v*approaches

*c*from 0.9

*c*.

### 4.2. Dielectric constant

*r*and

*N*kept fixed at

*r*= 0.5

*a*and

*N*= 21, let us examine how the SPR spectrum varies as the dielectric constant e of the cylinders changes in the monolayer. We select three values of

*ε, ε*= 4.41, 1 - (

*ω*

_{p}/

*ω*)

^{2}and - ∞. The first case corresponds to the dielectric constant of fused quartz with e nearly twice as large as that used above, the second is the dielectric constant of a Drude metal with

*ω*

_{p}the plasma frequency, and the third is the dielectric constant of a perfect conductor. To avoid the poor convergence of the cylindrical-wave expansion for the metallic cylinders in contact, we created a narrow opening between the cylinders by setting

*r*= 0.45

*a*in the Drude case. We assumed

*ω*

_{p}

*a*/2

*πc*= 1, i.e., the plasma wavelength equals the lattice constant. Calculation is made for the monolayer system using

*v*= 0.99999

*c*and

*b*= 3.33

*a*, as before.

*N*= ∞ system. Panels (a) and (b) depict the result of dielectric cylinders, panels (c) and (d) treat the Drude cylinders, and panel (e) presents the spectrum of the cylinders of a perfect conductor. Panels (b) and (d) also involve the band structures of the monolayer. Considering

*v*is ultra-relativistic, we only plotted the even-parity PhB structure. Note that for the perfect conductor case of panel (e), the ODOS does not present any peaks except for Wood’s anomaly and the PhB structure is completely absent.

29. V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B **60**, 5359–5365 (1999). [CrossRef]

30. H. van der Lem and A. Moroz, “Towards two-dimensional complete photonic bandgap structures below infrared wavelengths,” J. Opt. A **2**, 395–399 (2000). [CrossRef]

31. T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B **64**, 045117 (2001). [CrossRef]

*ωa*/2

*πc*= 0.5, which has a modest group velocity, is strongly coupled to the evanescent wave, yielding a very strong SPR signal. We thus conclude that uSPR carries information of PhBs of SPP origin.

*v*lines. This reflects the absence of the PhB structure in the array of perfect-conductor cylinders. Thus, we can conclude that the uSPR is peculiar to dielectric and metallic PhCs with finite dielectric function and is completely absent in the systems without PhBs.

### 4.3. Number of cylinders

*N*= 21 or

*N*= ∞. At small N, typically less than 8, the PhB structure is not clearly visible in the intensity map of the SPR on the (

*k*

_{x},

*ω*) plane. On the other hand, at large

*N*the PhB structure is clearly visible as demonstrated in Figs. 5 and 6. In this region, however, change of the SPR intensity map with increasing

*N*is less remarkable. Nevertheless, if we have a close look at the spectral line shapes of the uSPR, they indeed change with

*N*. To investigate this feature, we consider the SPR spectra at a fixed solid angle (

*θ,ϕ*) = (60°, 180°) as a function of frequency. Figure 11 shows the spectra for various

*N*. The SPR signals are strongly enhanced around (

*ωa*/2

*πc*= 1.195, which corresponds to an intersection point between the line of

*k*

_{x}-

*h*= (

*ω*/

*c*)cosθ (see Eq. (3)) and the PhB structure

*ω*=

*ω*

_{n}(

*k*

_{x},0). The intersection point is off the shifted

*v*lines and thus is indeed an uSPR signal. We can clearly observe that as

*N*increases, the intensity at the peak grows but seems to be saturated to a certain value. This implies that the radiation intensity of the uSPR per unit length of the electron trajectory decreases at large

*N*and eventually vanishes at

*N*= ∞. This property is reasonable because the uSPR is completely forbidden in the system of perfect translational invariance with

*N*= ∞. On the other hand, we also found that the intensity of the cSPR on the shifted

*v*lines increases almost linearly with

*N*, as expected from the conventional theory of the SPR. Therefore, at very large

*N*the cSPR signals will dominate over the uSPR ones. However, even at

*N*= 200 we found that the SPR intensity map does not differ so much from Fig. 5, in which the uSPR signals are rather stronger than the cSPR ones. Besides, in Fig. 11 we can clearly observe that the spectral width of the peak decreases with increasing

*N*. This property reflects better confinement of the radiation energy for larger

*N*. This width should converge to a certain value at

*N*= ∞, which is inversely proportional to the life-time of the relevant photonic band mode. This is nothing but the homogeneous broadening of the spectral line width of the uSPR.

### 4.4. Number of stacking layers

*y*direction. As we increase the number

*N*

_{l}of the stacking layers, the ODOS reveals a progressively finer structure as a function of frequency. Each peak of ODOS corresponds to a quasi-guided PhB mode confined in the stacked layers. The typical peak-to-peak distance in frequency is inversely proportional to

*N*

_{l}. Moreover, each peak is generally getting sharper.

*N*

_{l}is large enough, the scattering of the evanescent wave in the ultra-relativistic regime is identical to the transmission and reflection of a TE-polarized plane wave that enters the PhC with its left edge as an entrance surface. The slab PhC in question has a finite thickness

*N*

_{a}in the

*x*direction and has a large extension in the

*y*direction with the entrance surface parallel to the

*y*

_{z}plane. The wave vector component

*k*

_{∥}parallel to the entrance surface is conserved, and the incident plane wave excites the bulk PhB modes having the same

*k*

_{∥}. There is no momentum conservation in the

*x*component; in principle the light excites any PhB modes of arbitrary

*k*

_{x}.

*k*

_{z}) plays the role of

*k*

_{∥}in the above analogy, and thus we may set

*k*

_{∥}≃ 0 in the ultra-relativistic case, provided

*k*

_{z}= 0. Accordingly, the incident plane wave has the wave vectors (

*k*

_{x}, 0,0) and propagates in the Γ-

*X*direction of the square lattice. The bulk PhB modes along Γ-

*X*are thus excited. The conclusion is thus that uSPR will carry information along Γ-

*X*of the bulk PhB modes if both

*N*and

*N*

_{l}are sufficiently large. According to the above arguments, the SPR intensity is expected to be enhanced in the forward and backward directions, which correspond to the specular transmission and reflection. In addition, if

*ωa*/2

*πc*> 1, the SPR intensity is expected to be enhanced also on the curves

*h*= 2

*πn*/

*a*(

*n*: integer) along the

*k*

_{y}axis.

*ε*= 2.05,

*v*= 0.99999

*c*and

*b*= 3.33

*a*. In Fig. 12(a) the spectrum from the double-layer (

*N*

_{l}= 2) structure with

*N*= 21 is shown. The intensity map overlaid with the PhB structure of the double layers (but for

*N*= ∞) is shown in panel (b). As before, we plotted only the even-parity PhB structure. In the double layer, the mirror plane lies midway between the layers. We see the number of bands is almost twice that of the monolayer band structure shown in Fig. 3. This is reasonable, since the degenerate band-structures of each of the monolayers are split in the double layer. Obviously there is a very good correlation of the strong signals of uSPR with the band structure of the even parity.

*N*

_{l}= 20 and

*N*= 8. We consider this to be a test system simulating the slab-type PhC of square lattices. We observe at once a signal of high intensity along a hyperbolic curve whose bottom is found at (

*k*

_{x}

*a*/2

*π*,

*ωa*/2

*πc*) = (0,1). Obviously, this curve corresponds to Eq. (9) with

*h*=1. Strong SPR signals other than the hyperbolic curve are found at

*ωa*/2

*πc*= 0.73, 0.93, and 1.46. To identify these signals, Fig. 12(c) was overlaid with the even-parity PhB structure along the Γ -

*X*direction of the square lattice. The result is shown in Fig. 12(d). As can be clearly seen, the strong signals correspond to the anti-crossing points of the even-parity PhB structure. The bright curve connected to the strong signal around

*ωa*/2

*ωc*= 0.73 is shown to be along the PhB dispersion curve. Thus, we can conclude that the intensity map of the uSPR correlates well with the corresponding PhB structure even in the case of stacked monolayers.

### 4.5. Azimuthal angle

*k*

_{z}= 0 (

*ϕ*= 0° and 180°), that is, we have examined the radiation emitted within the

*x*

_{y}plane. We here investigate the

*ϕ*dependence. For this purpose, we write the differential cross section of SPR in polar coordinates [19

**69**, 125106 (2004). [CrossRef]

*z*=

*z*

_{0}.

*k*

_{z}is generally small compared with that of

*k*

_{z}= 0, and

*k*

_{z}is a conserved quantity in the scattering by the PhC. Therefore, the

*k*

_{z}dependence of the observed SPR will be controlled dominantly by that of the decaying exponential exp(-|Γ|

*b*) of the initial light. This exponential decreases with increasing |

*k*

_{z}|, so that the radiation is dominated by the SPR of

*k*

_{z}= 0.

*θ,ϕ*). In the ultra-relativistic regime it follows that

*ω*and

*ϕ*(≠0°, 180°) is dominated in the forward (

*θ*= 0°) and backward (

*θ*= 180°) directions. Similarly, at a given

*θ*, the SPR cross section is dominated around the plane perpendicular (

*θ*=0° and 180°) to the cylindrical axis.

## 5. Summary and discussions

*k*

_{z}= 0, so that the evanescent wave can be regarded as a plane wave propagating in the direction of the trajectory. This yields a peculiar radiation emission from the PhCs, which cannot be explained by the conventional theory of the SPR in which the finiteness of PhC is treated as infinite. The spectrum of the uSPR can be used as a probe of the PhB structure of the quasi-guided modes having the even-parity symmetry with respect to the relevant mirror plane.

*c*and 0.99

*c*. We also found that the uSPR is completely absent in the perfect-conductor cylinders because of the absence of PhBs. Otherwise, the spectra of the uSPR correlate with the corresponding PhB structure very well. We also found that the cross section of the SPR at an ultra-relativistic velocity is highly directive within the plane normal to the cylindrical axis.

## Acknowledgments

## References and links

1. | S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. |

2. | V. P. Shestopalov, |

3. | J. M. Wachtel, “Free-electron lasers using the Smith-Purcell effect,” J. Appl. Phys. |

4. | 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 |

5. | A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation-pattern of Smith-Purcell radiation,” J. Opt. Soc. Am. B |

6. | I. Shih, W. W. Salisbury, D. L. Masters, and D. B. Chang, “Measurements of Smith-Purcell radiation,” J. Opt. Soc. Am. B |

7. | 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. |

8. | 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 |

9. | P. M. van den Berg, “Smith-Purcell radiation from a point charge moving parallel to a reflection grating,” J. Opt. Soc. Am. |

10. | 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 |

11. | Y. Shibata, S. Hasebe, K. Ishi, S. Ono, M. Ikezawa, T. Nakazato, M. Oyamada, S. Urasawa, T. Takahashi, T. Mat-suyama, 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 |

12. | J. H. Brownell, J. Walsh, and G. Doucas, “Spontaneous Smith-Purcell radiation described through induced surface currents,” Phys. Rev. E |

13. | R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. |

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

15. | F. J. Garcìa de Abajo, “Smith-Purcell radiation emission in aligned nanoparticles,” Phys. Rev. E |

16. | K. Ohtaka and S. Yamaguti, “Theoretical study of the Smith-Purcell effect involving photonic crystals,” Opt. Spectrosc. |

17. | S. Yamaguti, J. Inoue, O. Haeberlé, and K. Ohtaka, “Photonic crystals versus diffraction gratings in Smith-Purcell radiation,” Phys. Rev. B |

18. | F. J. Garcìa de Abajo and L. A. Blanco, “Electron energy loss and induced photon emission in photonic crystals,” Phys. Rev. B |

19. | T. Ochiai and K. Ohtaka, “Relativistic electron energy loss and induced radiation emission in two-dimensional metallic photonic crystals. I. Formalism and surface plasmon polariton,” Phys. Rev. B |

20. | 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 |

21. | 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 |

22. | N. Horiuchi, T. Ochiai, J. Inoue, Y. Segawa, Y. Shibata, K. Ishi, Y. Kondo, M. Kanbe, H. Miyazaki, F. Hinode, S. Yamaguti, and K. Ohtaka, “Exotic radiation from a photonic crystal excited by an ultra-relativistic electron beam,” cond-mat/0604624. |

23. | F. J. Garcìa 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. |

24. | F. J. Garcìa 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 |

25. | C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in photonic crystals,” Science |

26. | A. S. Kesar, M. Hess, S. E. Korbly, and R. J. Temkin, “Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating,” Phys. Rev. E |

27. | K. Sakoda, |

28. | K. Ohtaka, J. Inoue, and S. Yamaguti, “Derivation of the density of states of leaky photonic bands,” Phys. Rev. B |

29. | V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B |

30. | H. van der Lem and A. Moroz, “Towards two-dimensional complete photonic bandgap structures below infrared wavelengths,” J. Opt. A |

31. | T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B |

32. | M. A. Kumakhov and G. Shirner, |

33. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(230.3990) Optical devices : Micro-optical devices

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 9, 2006

Revised Manuscript: July 28, 2006

Manuscript Accepted: July 28, 2006

Published: August 7, 2006

**Citation**

Tetsuyuki Ochiai and Kazuo Ohtaka, "Theory of unconventional Smith-Purcell radiation in finite-size
photonic crystals," Opt. Express **14**, 7378-7397 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7378

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### References

- S. J. Smith and E. M. Purcell, "Visible light from localized surface charges moving across a grating," Phys. Rev. 92, 1069 (1953). [CrossRef]
- V. P. Shestopalov, The Smith-Purcell effect (Nova Science, New York, 1998).
- J. M. Wachtel, "Free-electron lasers using the Smith-Purcell effect," J. Appl. Phys. 50, 49-56 (1979). [CrossRef]
- 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]
- A. Gover, P. Dvorkis, and U. Elisha, "Angular radiation-pattern of Smith-Purcell radiation," J. Opt. Soc. Am. B 1, 723-728 (1984). [CrossRef]
- I. Shih, W. W. Salisbury, D. L. Masters, and D. B. Chang, "Measurements of Smith-Purcell radiation," J. Opt. Soc. Am. B 7, 345-350 (1990). [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]
- 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]
- P. M. van den Berg, "Smith-Purcell radiation from a point charge moving parallel to a reflection grating," J. Opt. Soc. Am. 63, 1588-1597 (1973). [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]
- 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]
- J. H. Brownell, J. Walsh, and G. Doucas, "Spontaneous Smith-Purcell radiation described through induced surface currents," Phys. Rev. E 57, 1075-1080 (1998). [CrossRef]
- R. W. Wood, "Anomalous Diffraction Gratings," Phys. Rev. 48, 928-936 (1935). [CrossRef]
- J. B. Pendry and L. Martín-Moreno, "Energy-loss by charged-particles in complex media," Phys. Rev. B 50, 5062-5073 (1994). [CrossRef]
- F. J. García de Abajo, "Smith-Purcell radiation emission in aligned nanoparticles," Phys. Rev. E 61, 5743-5752 (2000). [CrossRef]
- K. Ohtaka and S. Yamaguti, "Theoretical study of the Smith-Purcell effect involving photonic crystals," Opt. Spectrosc. 91, 477-483 (2001). [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]
- 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]
- 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]
- 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(R) (2004). [CrossRef]
- N. Horiuchi, T. Ochiai, J. Inoue, Y. Segawa, Y. Shibata, K. Ishi, Y. Kondo, M. Kanbe, H. Miyazaki, F. Hinode, S. Yamaguti, and K. Ohtaka, "Exotic radiation from a photonic crystal excited by an ultra-relativistic electron beam," cond-mat/0604624.
- 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]
- 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]
- C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, "Cerenkov radiation in photonic crystals," Science 299, 368-371 (2003). [CrossRef] [PubMed]
- A. S. Kesar, M. Hess, S. E. Korbly, and R. J. Temkin, "Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating," Phys. Rev. E 71, 016501 (2005). [CrossRef]
- K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
- K. Ohtaka, J. Inoue, and S. Yamaguti, "Derivation of the density of states of leaky photonic bands," Phys. Rev. B 70, 035109 (2004). [CrossRef]
- V. Yannopapas, A. Modinos, and N. Stefanou, "Optical properties of metallodielectric photonic crystals," Phys. Rev. B 60, 5359-5365 (1999). [CrossRef]
- H. van der Lem and A. Moroz, "Towards two-dimensional complete photonic bandgap structures below infrared wavelengths," J. Opt. A 2, 395-399 (2000). [CrossRef]
- T. Ito and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001). [CrossRef]
- M. A. Kumakhov and G. Shirner, Atomic Collisions in Crystals (Gordon and Breach Science Publishers, New York, 1989).
- L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, p. 408 (Butterworth- Heinemann, Oxford, 1985).

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