Unified model of electron beam shot noise and coherent spontaneous emission in the helical wiggler free electron laser

A numerical algorithm for simulating electron beam shot noise in free electron lasers (FELs) is presented. Shot noise is a source of spontaneous emission that may be amplified in the self-amplified spontaneous emission regime of operation. This regime is of great importance to XUV and x-ray FELs where the spontaneous emission is currently the only effective source available for amplification. The algorithm uses a quasiuniform phase-space distribution of appropriately charge weighted macroparticles. The statistical properties of the macroparticles are derived directly from the temporal Poisson statistical properties of the real electron distribution. Unlike previous algorithms, ours does not rely upon any averaging over a resonant radiation period time scale and so more correctly describes the underlying physics. The algorithm also allows shot noise to be modeled self-consistently in unaveraged FEL models which are able to describe subwavelength phenomena such as coherent spontaneous emission (CSE). The algorithm is used in the unaveraged 1D FEL numerical simulation code FemFel and demonstrates spontaneous emission due to shot noise and CSE in both rectangular and Gaussian electron pulse current profiles. The preliminary results show good qualitative and quantitative agreement with theory.

Figure 1

The probability distribution $P(|b|)$ for the ensemble of ${10}^5$ macroparticle distributions as a histogram. The solid line is the Rayleigh distribution function of mean equal to 1.

Figure 2

The probability distribution $P(|b{|}^2)$ for the ensemble of ${10}^5$ macroparticle distributions as a histogram. The solid line is the negative exponential distribution function of mean equal to 1.

Figure 3

Schematic of macroparticle loading algorithm. The bottom left three grid boxes show the notation used for the grid dimensions and center. The randomness in the macroparticle charge via the Poisson random variate $N_j$ is demonstrated via their different relative sizes. The noise due to the random positioning is demonstrated via the variation about the grid centers.

Figure 4

Top panel: The scaled intensity, $|A{|}^2$, as a function of scaled pulse position ${\overline{z}}_1$ for a scaled distance through the interaction region of $\overline{z}=5$ and a rectangular profile electron pulse duration of ${\overline{l}}_e=6$ between $0<{\overline{z}}_1<6$. The slippage region is $0<{\overline{z}}_1<5$; the steady-state region is $5<{\overline{z}}_1<6$; the vacuum region is ${\overline{z}}_1>6$. Bottom panel: The scaled PSD, $P$, of the radiation as a function of the scaled frequency $f$.

Figure 5

Top panel: The filtered, scaled intensity, $|A{|}^2$, as a function of scaled pulse position ${\overline{z}}_1$ for a scaled distance through the interaction region of $\overline{z}=5$ and a rectangular profile electron pulse duration of ${\overline{l}}_e=6$ between $0<{\overline{z}}_1<6$ . The slippage region is $0<{\overline{z}}_1<5$; the steady-state region is $5<{\overline{z}}_1<6$; the vacuum region is ${\overline{z}}_1>6$. Bottom panel: The filtered scaled PSD, $P$, of the radiation as a function of the scaled frequency $f$.

Figure 6

The filtered scaled average radiation intensity $⟨|A{|}^2⟩$ (◇) and modulus of the bunching parameter $|b|$ (+) over the single period interval $2.5<{\overline{z}}_1<2.6$ of a rectangular profile electron pulse of ${\overline{l}}_e=6$. For $\overline{z}<2.5$ the interval is in the steady-state region, and for $\overline{z}>2.5$ the slippage region.

Figure 7

Top panel: The scaled intensity, $|A{|}^2$, as a function of scaled pulse position ${\overline{z}}_1$ for a scaled distance through the interaction region of $\overline{z}=2$ and a Gaussian profile electron pulse of total duration ${\overline{l}}_e=18$ and ${\sigma }_e=3$. Bottom panel: The scaled PSD, $P$, of the radiation as a function of the scaled frequency $f$.

Figure 8

Top panel: The filtered scaled intensity, $|A{|}^2$, as a function of scaled pulse position ${\overline{z}}_1$ for a scaled distance through the interaction region of $\overline{z}=2$ and a Gaussian profile electron pulse of total duration ${\overline{l}}_e=18$ and ${\sigma }_e=3$. Bottom panel: The filtered scaled PSD, $P$, of the radiation as a function of the scaled frequency $f$.

Figure 9

The filtered scaled average radiation intensity $⟨|A{|}^2⟩$ (◇) and modulus of the bunching parameter $|b|$ (+) over the single period interval $8.95<{\overline{z}}_1<9.05$ of a Gaussian profile electron pulse of ${\overline{l}}_e=18$ and ${\sigma }_e=3$.