Advection-Induced Spectrotemporal Defects in a Free-Electron Laser

We evidence numerically and experimentally that advection can induce spectrotemporal defects in a system presenting a localized structure. Those defects in the spectrum are associated with the breakings induced by the drift of the localized solution. The results are based on simulations and experiments performed on the super-ACO free-electron laser. However, we show that this instability can be generalized using a real Ginzburg-Landau equation with (i) advection and (ii) a finite-size supercritical region.

Figure 1

(a) Experimental setup for a storage-ring FEL. The wave is constituted of the optical pulse (several picoseconds duration) that experiences round-trips between two mirrors $M_1$ and $M_2$. At each passage the pulse is amplified by the interaction with the relativistic electron bunch of a storage ring (several tens of picoseconds, energy in the GeV domain) in an optical klystron (OK), see Refs. 22, 23 for details. (b) Illustration of the theoretical framework: One considers the complex envelope profile $e_n(\theta )$ of the field pattern at each electron passage, and takes the continuous limit $n\to T$.

Figure 2

Numerical solution of the FEL model (1–3) versus the drift parameter $v$. The first line (a),(d),(g),(j) represents the pulse shapes versus time $|e(\theta ,T){|}^2$. The evolutions of the spatial Fourier transforms $\tilde{e}(k,T)$ are represented below each regime: The second and third lines (b),(e),(h),(k) and (c),(f),(i),(l) represent the associated amplitudes $|\tilde{e}(k,T){|}^2$ and phases $\arg [\tilde{e}(k,T)]$ associated with (a),(d),(g),(j), respectively. The drift velocity increases from left to right: First column (a),(b),(c): $v=0$, below the instability threshold; second column (d),(e),(f): $v=0.5$, moderately above the threshold; third column (g),(h),(i): $v=0.7$; fourth column (j),(k),(l): $v=3.4$, a typical noise-sustained structure observed at high drift velocity. $T=1000$ corresponds to 20 ms, and $\theta =100$ to 8.3 ps. Note that the holes present in the spectrum evolutions are associated with zeroes of the field.

Figure 3

Experimental results. (a)–(d): Pulse shape evolutions $|e(\theta ,T){|}^2$ recorded with a streak camera. (e)–(g): Optical spectrum evolutions $|\tilde{e}(k,T){|}^2$ associated with (b)–(d) (the parameters are identical, but spectra and streak recordings are not synchronized). (a): case of rf frequency ${\nu }_0$ associated to almost perfect synchronism (i.e., $v\approx 0$); (b),(e): case of rf frequency slightly just beyond the lower instability threshold (${\nu }_0-0.5\mathrm{Hz}$); (c),(f): case of rf frequency well beyond the upper instability threshold (${\nu }_0+1.5\mathrm{Hz}$); (d),(g): noise-sustained structure observed far from the instability threshold (${\nu }_0+3\mathrm{Hz}$).

Figure 4

Numerical integration of the Ginzburg-Landau equation with ramps and advection (4). (a)–(c): case of global coupling [$S$ given by Eq. (6)], $v=0.6$, $ϵ=.0012$, $R=4$. (d)–(f): case of local coupling [$S$ given by Eq. (5)], $v=0.6$, $ϵ=0.01$, $R=0.04$. (a),(d) represent $|e(z,T)|$, (b),(e): $|\tilde{e}(k,T)|$, (c),(f): $\arg [\tilde{e}(k,T)]$.