2. Numerical simulation

and communications technology. Optical techniques for a generation and detection of the THz radiation usually require ultrafast short-pulsed lasers. As an alternative way, vacuum electronicbased techniques have attracted much attention to develop next-generation table-top THz radiation sources [3]. It has been known as Smith-Purcell radiation (SPR) since the 1950s that EM radiation can be obtained by passing electron beam (e-beam) accelerated at a relativistic speed along the surface of periodically corrugated metallic grating [4]. The wavelength λSPR and the radiation angle θSPR of SPR measured from the direction of the e-beam satisfy the following

1

where Λ is the period of the grating, βc is the electron velocity, c is the speed of light in vacuum, and integer n is the order. Therefore, one can choose any spectral range of EM radiation in principle by appropriately designing the grating period Λ. The original type of SPR is not efficient enough to be widely utilized; however, there has been renewed interest in this area of research since the observation of superradiance by Urata et al. in 1998 [5]. They used an electron gun in a scanning electron microscope to flow a large current and discovered a nonlinearly growing radiation power (superradiance). Theoretical [6, 7] and numerical [8–10] investigations revealed that the bunching of an e-beam due to the interaction with induced surface waves on the grating was essential to achieve superradiance through intrabunch and interbunch double coherence. In such numerical investigations, the particle-in-cell finitedifference time-domain (PIC-FDTD) method has been widely employed to reproduce the

On the other hand, quite significant progress has been made in the researches on metamaterials in recent years, and various novel optical effects have been proposed and demonstrated, such as negative refraction, superlensing, and optical cloakings [11–13]. Based on metamaterials' concept, one can design a rich variety of optical materials with unique dispersion characters which cannot be obtained in nature. The metamaterials' concept also offers new designing freedom of surface waves. It has been believed that surface waves like surface plasmon polaritons (SPPs) in the visible or near infrared cannot be supported in longer wavelength range like in THz because metals tend to behave as a perfect electric conductor (PEC). However, Pendry et al. showed that surface waves like SPPs could be supported even on PECs provided that there were arrays of corrugations or holes on metals [14, 15]. The dispersion relations of such surface waves resemble those of SPPs, and the surface waves introduced by Pendry et al. are usually called spoof SPPs. Dispersion relations of the spoof SPPs strongly depend on the dimensions of corrugations and holes, which in turn implies that almost arbitrary dispersion relations for the spoof SPPs can be designed through an appropriate choice of structural parameters. Based on this concept, Gan et al. demonstrated an ultrawide-bandwidth slow-light system with a graded metallic grating with linearly graded depths [16]. Since the dispersion of the spoof SPPs is strongly dependent on the geometrical parameters of the grooves, the upper limit (cutoff) frequencies for the existence of spoof SPPs are also determined by the groove geometries. The group velocity of each mode approaches zero near the cutoff frequency; therefore, graded grating structures with spatially varying dispersions are capable of stopping spoof SPPs with different frequencies at different locations

<sup>β</sup> � cos <sup>θ</sup>SPR 

(1)

<sup>λ</sup>SPR <sup>¼</sup> <sup>Λ</sup> j j n

simple relation:

114 Metamaterials and Metasurfaces

Smith-Purcell superradiance.

In this section, numerical simulation techniques employed in this study, simplified PIC-FDTD method, is described in detail, and parameters for our simulations are summarized.
