Spatial Growth Rate Outperforms Interaction Impedance in Predicting Smith-Purcell Radiation Gain

Smith-Purcell radiation (SPR) occurs when charged particles travel near a periodic structure [1]. In this study, we investigate the shape of the SPR dispersion curves on SPR growth rate, and start current requirements. Our cold- and hot-tube dispersion analysis [2] indicates that a groove with a higher aspect ratio (height by period of the grating) produces a flatter dispersion curve, exhibits higher spatial growth rates, and, consequently, a lower start current, particularly near the top band edge. The coupling impedance [2], a common parameter in radio frequency (RF) devices that measures the electromagnetic interaction between a beam and its surrounding structure, likewise increases with a higher aspect ratio of the groove or flatter dispersion. However, interaction impedance and spatial growth rate exhibit opposite trends in phase spaces, with higher interaction impedance in the fundamental phase space but higher spatial growth rates in the negative phase space. Starting current requirements calculated using particle-in-cell (PIC) simulations confirm the correlation between spatial growth rate and starting current. Specifically, lower starting currents in the negative phase space and higher in the fundamental phase space reflect the same trend as spatial growth rate, validating our hot-tube analysis. Therefore, our hot-tube dispersion offers a more accurate approach than coupling impedance for predicting SPR starting current, enabling grating optimization and radiation enhancement.