Simulation of Smith-Purcell radiation using a particle-in-cell code

A simulation of the generation of Smith-Purcell (SP) radiation at microwave frequencies is performed using the two-dimensional particle-in-cell code MAGIC. The simulation supposes that a continuous, thin (but infinitely wide), monoenergetic electron beam passes over a diffraction grating, while a strong axial magnetic field constrains the electrons to essentially one-dimensional motion. The code computes the time-dependent electric and magnetic fields by solving the Maxwell equations using a finite element approach. We find that the passage of the beam excites an evanescent electromagnetic wave in the proximity of the grating, which in turn leads to bunching of the initially continuous electron beam. The frequency and wave number of the bunching are determined, and found to be close to those proposed by Brau and co-workers in recent work. This frequency is below the threshold for SP radiation. However, the bunching is sufficiently strong that higher harmonics are clearly visible in the beam current. These harmonic frequencies correspond to allowed SP radiation, and we see strong emission of such radiation at the appropriate angles in our simulation, again in agreement with Brau’s predictions. We also find that at the ends of the grating, some of the evanescent wave is diffracted away from the surface, and radiation below the threshold occurs. In addition, we observe a second evanescent wave at the same frequency, but with a different wave number. The existence of this wave is also predicted by the theory, although its presence in our simulation is unexpected. Numerical estimates of the growth of the evanescent wave are also in reasonable agreement with the predictions, although the precise form of the dependence of the gain on beam current remains hard to establish.

Figure 1

Geometry used in MAGIC to calculate SP radiation.

Figure 2

Dispersion relation for our grating and intersection with the beam line. The operating point $P$ is shown, as well as a second point $P^{\prime }$ which has the same frequency but $k^{\prime }=K\mathrm{\text{−}}k$. The open squares indicate the results of a MAGIC simulation in which a fixed frequency current source is placed in a groove and excites the evanescent waves. The two wave numbers corresponding to each frequency are found by FFT.

Figure 3

$x\mathrm{\text{−}}y$ contour map of $B_z$ obtained at 34 ns.

Figure 4

Time signal of $B_z$ and corresponding FFT from a detector placed at 64°.

Figure 5

Amplitude of the major FFT coefficients as a function of angle. Also shown are the first and second order SP wavelengths as a function of angle, together with the wavelengths of the fundamental (evanescent), second, and third harmonics.

Figure 6

(a) Current as a function of $x$ at three different times. (b) Current as a function of time at three different locations along the grating.

Figure 7

Projected phase-space distributions: (a) density of electrons in the $x\mathrm{\text{−}}y$ plane at 32 ns. Bunching is evident. (b) kinetic energy-$x$ density at same time.

Figure 8

(a) $B_z$ ($x$) near the grating at three different times, showing an approximate period of 4.8 cm, and indicating a slow movement in the negative $x$ direction. (b) FFT of $B_z$ ($x$) (solid curve, with two peaks) compared to FFT of current (dashed curve, one peak).

Figure 9

(a) Temporal gain $\mathrm{Im}(2\omega )$ as a function of current (b) CSP frequency as a function of current.