1. Introduction

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

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 [2–4]. 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]. There have been several studies of superradiant and coherent SPR [6–8]. H. L. Andrews et al. developed a new theory on superradiant SPR [7–8]. 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.

There have been a number of experiments in the millimeter to visible spectra and theoretical analysis up to terahertz-order SPR[9–10, 12–14]. Ochiai and Ohtaka investigated SPR from photonic crystals in the optical region by using a scattering matrix method based on plane-wave expansion [15, 16]. However, there has been little theoretical research on SPR from metal gratings in the optical region. We studied the problems related to the behavior of and simulated SPR in the optical region using a two-dimensional finite-difference time-domain (FDTD) method.

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.

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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

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

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

where N ₀ is the constant electron density, and $\sigma \left(=\frac{w}{2\sqrt{2\ln \left(2\right)}}\right)$ 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

and

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

is substituted into Maxwell’s equation.

The parameters used for simulations are summarized in Table 1.

Table 1. Main parameters for the simulation.

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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 H_z 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 H_z observed at observation point with θ=-10°. (b) FFT spectrum.

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Fig. 3. FFT spectrum for H_z (a) before and (b) after 86 fs.

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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]. 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].

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Figure 5 shows the FFT spectrum for H_z as a function of the observation angle θ. The symbols (a) to (c) correspond to those in Figs. 2(b) and 3. Solid lines (b) and (b’) indicate the SPR wavelengths calculated using Eq. (1) with n=1 and 2, respectively. Good agreement is obtained between the FDTD simulation and Eq. (1). The spectrum peaks (a) and (c) will be discussed in the next section.

 

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

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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]. 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

where ε_r and ε_d 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.

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As shown in the previous section, the spectrum component (c) is caused by mimic-SPP and emitted after the electron bunch has passed the grating region. Andrews et al. reported mimic-SPP [7–8]. According to their theory, the dispersion characteristics of the mimic-SPP are given by

where k is the wave number in the x-axis direction, δ ₀₀=1, $K=\frac{2\pi }{l}$, ${\omega \prime }_{\mathrm{pl}}^2=\frac{n_e^{\prime }{q}^2}{{\epsilon }_0{m}_e}$, ${\kappa }_0^2=-\frac{{\omega }^2}{c^2}$, ${\chi }_p^{\prime }=\frac{-{\omega \prime }_{\mathrm{pl}}^2\sqrt{1-{\beta }^2}}{{\left[\omega -\beta c\left(k+\mathrm{pK}\right)\right]}^2}$, ${\alpha }_p^2={\left(k+\mathrm{pK}\right)}^2-\frac{{\omega }^2}{c^2}+\frac{{\omega \prime }_{\mathrm{pl}}^2}{c^2}$.

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. [7–8], 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]. 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.

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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.