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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 24 — Nov. 26, 2007
  • pp: 16222–16229
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Analysis of Smith-Purcell radiation in optical region

Simpei Taga, Koji Inafune, and Eiichi Sano  »View Author Affiliations


Optics Express, Vol. 15, Issue 24, pp. 16222-16229 (2007)
http://dx.doi.org/10.1364/OE.15.016222


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Abstract

Smith-Purcell radiation (SPR), emitted when an electron beam is traveling above a metallic grating, has attracted a lot of attention as a source of electromagnetic (EM) radiation in the millimeter to visible spectrum. We conducted a theoretical investigation of SPR in the optical region using a two-dimensional finite-difference time-domain (FDTD) method. The permittivity of metal was represented using the Drude model. During the simulation, we observed three types of EM radiations when an electron bunch passes above a metal grating. We think these three types of EM radiation were basic SPR, original surface plasmon polariton (SPP), and mimic-SPP, caused by the periodic grating structure. Our observations were in accordance with analytical models of original SPP and mimic-SPP EM radiation.

© 2007 Optical Society of America

1. Introduction

Smith-Purcell radiation (SPR) emitted from electrons moving above a metal grating [1

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

] has attracted a lot of attention. This is because SPR has possible applications in compact low-cost tunable high-power electromagnetic (EM) devices.

The wavelength λ of the radiation observed at the angle θ measured from the direction normal to a grating surface is given by

λ=ln(β1sinθ)
(1)

where l is the grating period, c the speed of light, βc the electron velocity, and n the order of the reflection from the grating (Fig. 1). Since the wavelength is set by the grating period, the SPR can produce EM fields from the millimeter-wave to the visible light range. This has advantages over other optical sources such as light-emitting diodes (LEDs) and semiconductor lasers because these optical sources require the use of an appropriate material with an energy bandgap corresponding to the desired wavelength in order to change the wavelength of the output. However, it is only necessary to change the grating period in a device that uses SPR in order to change the wavelength of the output, which makes such a device cheaper and easier to use.

Basic and incoherent SPR have been analyzed in many ways, including using diffraction theory and surface current models [2

2. P. M. van den Berg, “Smith-Purcell radiation from a line charge moving parallel to a reflection grating,” J. Opt. Soc. Am. 63, 689–698 (1973). [CrossRef]

4

4. 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]

]. In 1998, J. Urata et al. discovered a high-power SPR called superradiant SPR, which could not be explained by basic and incoherent SPR theory[5

5. J. Urata, M. Goldstein, M. F. Kimmitt, A. Naumov, C. Platt, and J. E. Walsh, “Superradiant Smith-Purcell emission,” Phys. Rev. Lett. 80, 516–519 (1998). [CrossRef]

]. There have been several studies of superradiant and coherent SPR [6

6. L. Schachter and A. Ron, “Smith-Purcell free-electron laser,” Phys Rev. A 40, 876–896 (1989). [CrossRef] [PubMed]

8

8. H. L. Andrews, C. A. Brau, and J. D. Jarvis, “Superradiant emission of Smith-Purcell radiation,” Phys. Rev. Special Topics 8, 110702 (2005).

]. H. L. Andrews et al. developed a new theory on superradiant SPR [7

7. H. L. Andrews and C. A. Brau, “Gain of a Smith-Purcell free-electron laser,” Phys. Rev. Special Topics 7, 070701 (2004).

8

8. H. L. Andrews, C. A. Brau, and J. D. Jarvis, “Superradiant emission of Smith-Purcell radiation,” Phys. Rev. Special Topics 8, 110702 (2005).

]. They used a continuous electron beam and evanescent wave to explain superradiant SPR in the following way: If a continuous electron beam that has a current that is sufficiently high travels above a grating, a wave in an evanescent mode traveling along the grating occurs in addition to the basic SPR. The group velocity of the evanescent wave can be either positive, as in a traveling-wave tube, or negative, as in a back-wave oscillator. The group velocity depends on the dispersive properties of the grating and the velocity of the electrons. At low electron beam energy, the evanescent wave travels backward and this causes nonlinear bunching of the electrons in the beam, which coherently enhances the basic SPR. This phenomenon and the low material cost of devices that use SPR means SPR may have applications in high-performance optical devices.

2. Description of the analysis

2.1 Geometry

We simulated the SPR using a FDTD method. FDTD methods have been successfully applied to various electromagnetic (EM) problems in the microwave to optical frequency regions. SPPs are p-polarized strongly localized surface waves formed at a material interface. Therefore, we used a transverse magnetic (TM)-FDTD simulation.

Figure 1 shows the geometry used for our simulations. The rectangular grating is formed from a silicon substrate coated by a thin film of Ag. The dielectric constant for Ag is assumed to be a complex number following the Drude model. The grating is placed at the center of the bottom of the simulation box and the grooves are uniform in the z direction. The entire region except for the grating is in air, which is enclosed by absorbers called the perfectly matched layer (PML). The whole simulation area is divided into a mesh with rectangle cells. The region (5.25<x≤7.35 µ m) surrounding the grating has a small cell size of dx=dy=1 nm, while the other region has a larger cell size of dx=dy=10 nm to reduce the computation time.

Fig.1. Geometry used for simulations.

2.2 Dielectric constant of the metal

Permittivity for a metal and a highly doped semiconductor depends on the incident EM radiation frequency. These materials are called dispersing media. In our simulation, we used the Drude model for Ag. The model is expressed by

εr(ω)=1+ωp2ω(jΓω)=1+χ(ω)
(2)
χ(ω)=ωp2ω(jΓω)
(3)

where ωp is the plasma frequency, and Γ the collision frequency. We used the parameter values so that the calculated εr matches the measured one [17

17. M. Moskovits, I. Srnova-Sloufova, and B. Vlckova, “Bimetallic Ag-Au nanoparticles: Extracting meaningful optical constants from the surface-plasmon extinction spectrum,” J. Chem. Phys. 116, 10435–10446 (2002). [CrossRef]

].

2.3 Electron bunch

We assumed the electron beam to be an electron bunch for simplicity in the simulations. The distribution of the electron density in the bunch is Gaussian and given by

ne=N0exp{(xx0)2+(yy0)22σ2}
(4)

where N 0 is the constant electron density, and σ(=w22ln(2)) is the variance of the bunch. The coordinate (x 0, y 0) is the moving center of the electron bunch and calculated using the momentum equations given by

dpdt=q(E+ν×B)
(5)

and

p=meν1ν2c2
(6)

Here p⃗ is the momentum, q the electron charge, E⃗ the electric field, v⃗ the velocity, B⃗ the magnetic flux density, and me the electron mass. The initial velocity was assumed to be βc. The current density

j=neqν
(7)

is substituted into Maxwell’s equation.

The parameters used for simulations are summarized in Table 1.

Table 1. Main parameters for the simulation.

table-icon
View This Table

3. Results of the simulation

In the basic SPR equation [Eq.(1)], the maximum wavelength obtainable for a given β and l, is λ max=l(1/β+1). For our case of E=30 keV, β=0.3, and l=200 nm, the maximum wavelength was calculated to be 783 nm.

Figure 2(a) shows the transient waveform of Hz observed at the observation point with θ=-10° and Fig. 2(b) shows the waveform’s fast Fourier transformation (FFT) spectrum. The transient waveform consists of waves with large amplitudes between 56 and 86 fs followed by waves with smaller amplitudes. The former waves were generated while the electron bunch passed above the grating, which will be shown later. We can see three large peaks denoted by (a), (b), and (c) in Fig. 2(b). To investigate the origins of these peaks, we calculated FFT spectra for the waveforms before and after 86 fs. Figures 3(a) and 3(b) show the results. The spectrum components (a) and (b) are radiated while the electron bunch passes above the grating, and the spectrum peak (c) is radiated after the electron bunch has passed the grating region.

Fig. 2. (a) Transient waveform of Hz observed at observation point with θ=-10°. (b) FFT spectrum.
Fig. 3. FFT spectrum for Hz (a) before and (b) after 86 fs.

Figures 4(a) and 4(b) are short films showing the magnetic field behaviors over two time intervals. In Fig. 4(a), the SPR is emitted when the electron bunch passes above the grating. We can easily determine the spectrum peak (b) for the SPR. In Fig. 4(b), a surface wave can be seen after the electron bunch has passed. This surface wave is caused by the periodic grating. We call this surface wave mimic surface-plasmon-polariton (mimic-SPP) after Pendry’s work to distinguish it from the original SPP[11

11. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

]. The spectrum peak (c) can be deduced for the mimic-SPP.

Fig. 4. (a) (2.31MB) Film of the magnetic field behavior from 40 to 49 fs [Media 1]. (b) (2.42MB) Film of the magnetic field behavior from 71 to 80 fs [Media 2].

Fig. 5. FFT spectrum for Hz as function of observation angle θ.

4. Discussion of the results

We believe that the spectrum peak (a) is caused by the original SPP. Generally, the SPP is localized at the interface between the air and metal. However, we can observe radiation from the SPP away from the interface. This is due to scattering from the corner of the grating as described by Cao [18

18. L. Cao, N. C. Panoiu, and R. M. Osgood Jr., “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007). [CrossRef]

]. We simulated the SPP behavior when electron bunches passed above a flat Ag surface by using an FDTD method. Figure 6 shows the induced SPP wavelength as a function of the bunch speed. The dispersion of the SPP is given by

ksp=ωc(εrεdεr+εd)12
(8)

where εr and εd are the permittivity of metal and air, respectively. The induced SPP wavelength is obtained from the intersection between the dispersion curve given by Eq. (8) and the beam line (ω=βck). The calculated SPP wavelength is shown as a solid line in Fig. 6. Good agreement is obtained between our FDTD simulation and simple calculation. Therefore, we can attribute the spectrum peak (a) observed in Fig. 2(b) to the SPP.

Fig. 6. Relationship between SPP wavelength and velocity of electron bunch.

R001+χ0S00=0
(9)
R00=tanh(κ0h)1+δ00p=11κ0aαpl4(k+pK)2a2×cos[(k+pK)a]1
(10)
S00=tanh(κ0h)1+δ00κ0aα0l4k2a2×cos[ka]1
(11)

where k is the wave number in the x-axis direction, δ 00=1, K=2πl, ωpl2=neq2ε0me, κ02=ω2c2, χp=ωpl21β2[ωβc(k+pK)]2, αp2=(k+pK)2ω2c2+ωpl2c2.

The frequency for induced mimic-SPP is obtained from the intersection between the dispersion curve calculated using Eqs. (9)(11) and the electron beam line. Figure 7(a) shows the calculation result. The frequency of induced mimic-SPP is independent of the observation angle. As pointed out by Andrews et al. [7

7. H. L. Andrews and C. A. Brau, “Gain of a Smith-Purcell free-electron laser,” Phys. Rev. Special Topics 7, 070701 (2004).

8

8. H. L. Andrews, C. A. Brau, and J. D. Jarvis, “Superradiant emission of Smith-Purcell radiation,” Phys. Rev. Special Topics 8, 110702 (2005).

], the induced mimic-SPP is a backward wave. A careful inspection of the short film shown in Fig. 4(b) reveals this backward propagation. Figure 7(b) shows the mimic-SPP wavelength obtained using FDTD simulations when the groove depth is varied. Andrews et al. developed a model that assumes the grating is a perfect electric conductor (PEC); therefore, we conducted simulations for PEC gratings. The results are shown in Fig. 7(b) along with the wavelengths calculated using the model. Good agreement is obtained between our FDTD simulations and the model.

We have described the behavior of two types of SPPs on the metal grating. The mimic-SPP as we call it here is also known as a localized surface plasmon (LSP). Cesario et al. demonstrated an enhanced transmission of light through a metal layer by coupling the LSP and the extended plasmon (which is the same as “the original SPP” as we call it here) [19

19. J. Cesario, M. U. Gonzalez, S. Cheylan, W. L. Barnes, S. Enoch, and R. Quidant, “Coupling localized and extended plasmons to improve the light extraction through metal films,” Opt. Express 15, 10533–10539 (2007). [CrossRef] [PubMed]

]. Unfortunately, in our simulation, no coupling between the original SPP and mimic-SPP (LSP) occurs in the radiation from the metal gratings. Coupling between SPP and LSP may result in different radiation characteristics. Investigation of the coupling effect in SPR will be an interesting field of research for the future.

Fig. 7. (a) D ispersion relation between M-SPP and electron bunch (l=200 nm, a=100n, h=200nm). (b) Groove depth dependence of mimic-SPP.

4. Conclusion

We have studied Smith-Purcell radiation in the optical region. We used the metal of a grating as a dispersive medium assuming the Drude model. Our simulation produced three kinds of EM radiation when an electron bunch passed above the metal grating. We think these EM radiations were basic SPR, original SPP and mimic-SPP, caused by the periodic grating structure. The results of our simulation were in accordance with analytical models of original SPP and mimic-SPP.

Acknowledgments

This work was financially supported in part by the Strategic Information and Communications R&D Promotion Programme (SCOPE) from the MIC, Japan, and by the Grant in Aid for Scientific Research (S) from the MEXT, Japan.

References and links

1.

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

2.

P. M. van den Berg, “Smith-Purcell radiation from a line charge moving parallel to a reflection grating,” J. Opt. Soc. Am. 63, 689–698 (1973). [CrossRef]

3.

P. M. van den Berg, “Smith-Purcell radiation from a point charge moving to a parallel reflection grating,” J. Opt. Soc. Am. 63, 1589–1597 (1973).

4.

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]

5.

J. Urata, M. Goldstein, M. F. Kimmitt, A. Naumov, C. Platt, and J. E. Walsh, “Superradiant Smith-Purcell emission,” Phys. Rev. Lett. 80, 516–519 (1998). [CrossRef]

6.

L. Schachter and A. Ron, “Smith-Purcell free-electron laser,” Phys Rev. A 40, 876–896 (1989). [CrossRef] [PubMed]

7.

H. L. Andrews and C. A. Brau, “Gain of a Smith-Purcell free-electron laser,” Phys. Rev. Special Topics 7, 070701 (2004).

8.

H. L. Andrews, C. A. Brau, and J. D. Jarvis, “Superradiant emission of Smith-Purcell radiation,” Phys. Rev. Special Topics 8, 110702 (2005).

9.

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,” Phy. Rev. E 57, 1061–1074 (1998). [CrossRef]

10.

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

11.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

12.

D. Li, Z. Yang, K. Imasaki, and G.-S. Park, “Particle-in-cell simulation of coherent and superradiant Smith-Purcell radiation,” Phys. Rev. Special Topics 9, 040701 (2006).

13.

J. T. Donohue, “Simulation of Smith-Purcell radiation using a particle-in-cell code,” Phys Pev. Special Topics 8, 060702 (2005).

14.

S. E. Korbly, A. S. Kesar, J. R. Sirigiri, and R. J. Temkin, “Observation of frequency-locked coherent terahertz Smith-Purcell radiation,” Phys. Rev. Lett. 94, 054803 (2005). [CrossRef] [PubMed]

15.

T. Ochiai and K. Ohtaka, “Electron energy loss and Smith-Purcell radiation in two- and three-dimentional photonic crystals,” Opt. Express 13, 7683–7697 (2005). [CrossRef] [PubMed]

16.

T. Ochiai and K. Ohtaka, “Relativistic electron energy loss and induced radiation emission in twodimensional metallic photonic crystals. II. photonic band effects,” Phys. Rev. B 69, 125107 (2004). [CrossRef]

17.

M. Moskovits, I. Srnova-Sloufova, and B. Vlckova, “Bimetallic Ag-Au nanoparticles: Extracting meaningful optical constants from the surface-plasmon extinction spectrum,” J. Chem. Phys. 116, 10435–10446 (2002). [CrossRef]

18.

L. Cao, N. C. Panoiu, and R. M. Osgood Jr., “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007). [CrossRef]

19.

J. Cesario, M. U. Gonzalez, S. Cheylan, W. L. Barnes, S. Enoch, and R. Quidant, “Coupling localized and extended plasmons to improve the light extraction through metal films,” Opt. Express 15, 10533–10539 (2007). [CrossRef] [PubMed]

OCIS Codes
(230.3990) Optical devices : Micro-optical devices
(230.6080) Optical devices : Sources
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: October 15, 2007
Revised Manuscript: November 20, 2007
Manuscript Accepted: November 20, 2007
Published: November 21, 2007

Citation
Simpei Taga, Koji Inafune, and Eiichi Sano, "Analysis of Smith-Purcell radiation in optical region," Opt. Express 15, 16222-16229 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-16222


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References

  1. S. J. Smith and E. M. Purcell, "Visible light from localized surface charges moving across a grating," Phys Rev. 92, 1069 (1953). [CrossRef]
  2. P. M. van den Berg, "Smith-Purcell radiation from a line charge moving parallel to a reflection grating," J. Opt. Soc. Am. 63, 689-698 (1973). [CrossRef]
  3. P. M. van den Berg, "Smith-Purcell radiation from a point charge moving to a parallel reflection grating," J. Opt. Soc. Am. 63, 1589-1597 (1973).
  4. 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]
  5. J. Urata, M. Goldstein, M. F. Kimmitt, A. Naumov, C. Platt, and J. E. Walsh, "Superradiant Smith-Purcell emission," Phys. Rev. Lett. 80, 516-519 (1998). [CrossRef]
  6. L. Schachter and A. Ron, "Smith-Purcell free-electron laser," Phys Rev. A 40, 876- 896 (1989). [CrossRef] [PubMed]
  7. H. L. Andrews, C. A. Brau, "Gain of a Smith-Purcell free-electron laser," Phys. Rev. Special Topics 7, 070701 (2004).
  8. H. L. Andrews, C. A. Brau, and J. D. Jarvis, "Superradiant emission of Smith-Purcell radiation," Phys. Rev. Special Topics 8, 110702 (2005).
  9. 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," Phy. Rev. E 57, 1061-1074 (1998). [CrossRef]
  10. I. Shih, W. W. Salisbury, D. L. Masters, and D. B. Chang, "Measurement of Smith-Purcell radiation," J. Opt. Soc. Am. B 7, 345-350 (1990). [CrossRef]
  11. J. B. Pendry, L. Martin-Moreno, F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
  12. D. Li, Z. Yang, K. Imasaki, and G.-S. Park, "Particle-in-cell simulation of coherent and superradiant Smith-Purcell radiation," Phys. Rev. Special Topics 9, 040701 (2006).
  13. J. T. Donohue, "Simulation of Smith-Purcell radiation using a particle-in-cell code," Phys Pev.Special Topics 8, 060702 (2005).
  14. S. E. Korbly, A. S. Kesar, J. R. Sirigiri, and R. J. Temkin, "Observation of frequency-locked coherent terahertz Smith-Purcell radiation," Phys. Rev. Lett. 94, 054803 (2005). [CrossRef] [PubMed]
  15. T. Ochiai and K. Ohtaka, "Electron energy loss and Smith-Purcell radiation in two- and three-dimentional photonic crystals," Opt. Express 13, 7683-7697 (2005). [CrossRef] [PubMed]
  16. 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]
  17. M. Moskovits, I. Srnova-Sloufova, B. Vlckova, "Bimetallic Ag-Au nanoparticles: Extracting meaningful optical constants from the surface-plasmon extinction spectrum," J. Chem. Phys. 116, 10435-10446 (2002). [CrossRef]
  18. L. Cao, N. C. Panoiu and R. M. OsgoodJr., "Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures," Phys. Rev. B 75, 205401 (2007). [CrossRef]
  19. J. Cesario, M. U. Gonzalez, S. Cheylan, W. L. Barnes, S. Enoch, and R. Quidant, "Coupling localized and extended plasmons to improve the light extraction through metal films," Opt. Express 15, 10533-10539 (2007). [CrossRef] [PubMed]

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