Watanabeet al.Reply:

Abstract

Watanabe et al. Reply: After carefully evaluating the preceding Comment by Bonafacio et al. [1] on our Letter [2], we concluded that those comments do not affect our experimental results, nor alter our analysis and the conclusions presented in our Letter. The following is the detailed response to the comments. In the following the detailed response to the comments are given in the order of remark (2), (3), and (1), followed by our conclusion.In our single-pass high-gain free-electron laser (FEL) amplifier experiments, the seed pulse is first amplified in the exponential gain regime where the pulse lengthens; after saturation of the exponential gain regime the FEL transitions into the superradiant regime. Superradiance in a FEL amplifier manifests itself in several ways in addition to the radiation power scaling as I 2 . In the superradiant regime the FEL amplifier pulse length shrinks, the spectral bandwidth increases, and the FEL pulse energy continues to grow (without the use of any undulator tapering) (Refs. [8, 11, 13, 18] of [2]). In our Letter the continuous growth of the FEL energy in the nonlinear regime (Fig. 2), the FEL pulse shortening by roughly 50% (Fig. 3), and the resulting spectral broadening (Fig. 3) were all observed and the results were consistent with the numerical simulations (Fig. 4). These simultaneous observations provide compelling proof of superradiance in our FEL amplifier experiments.The measurement of the power scaling as I 2 as suggested in remark (2) is very challenging for our experimental setup, as a change in the beam charge would affect the electron beam emittance, energy spread, bunch length, and the bunch current distribution, especially in the photoinjector and in the compressor. This means that the FEL saturated power and the distance where the FEL radiation evolves as a superradiant pulse are changed accordingly. For the above reasons, in our experimental conditions a simple scaling as P / I 2 is not expected. Note that measuring the FEL energy as a function of the undulator length has been widely used to characterize single-pass high-gain FELs [2 -5, 23].In [18] it is shown that after saturation the pulse propagates at a constant velocity approximately equal to c and scaling laws are derived in the variable z, defined as the bunch coordinate along the undulator. In response to remark (3) we show that it is straightforward to demonstrate under this assumption the equivalence with the scaling laws derived in Refs. [8, 12]. Here we indicate with (z l ; t l ) the time and position coordinates in the laboratory. In [18] the equations are expressed in a coordinate system (; z) moving at the constant velocity z c in the longitudinal direction.

Physics of Alfvén waves and energetic particles in burning plasmas

“…Equilibrium geometry and plasma nonuniformities influence the wave packet propagation speed and characteristic width as well. Because of its form, the intensity of the convectively amplified wave packet grows as the square of the distance; in analogy with the superradiance (Dicke, 1954) operation regime of a free electron laser (FEL), where the peak power also increases as the square of the distance along the undulator (Bonifacio et al, 1990(Bonifacio et al, , 1994Giannessi et al, 2005;Watanabe et al, 2007). The mechanism by which EPs eventually loose resonance by residual resonance detuning and are substituted by new resonant EPs reinforces this analogy (Zonca et al, 2015b).…”

Interference-induced peak splitting in extreme ultraviolet superfluorescence

“…In particular, superradiant amplification has been demonstrated in an underdense plasma [10]. The superradiance in a free-electron laser was observed too [11]. Furthermore, the work on SF indicates an important potential application for producing coherent, ultrashort, and intense laser pulses, particularly, in the high frequency regions [ultraviolet (UV), extreme ultraviolet (EUV), and x-ray] by a suitable choice of the system and pump wavelength.…”

High-Order-Harmonic Generation and Superradiance in a Seeded Free-Electron Laser

“…Coherent harmonics have been observed when seeding a single-pass FEL amplifier, and include the 3rd, 5th [19], and 7th harmonics [20]. The process of harmonic generation is expected to extend to higher orders when the FEL operates in the regime of super-radiance [21], in which a short optical pulse slips over the electron beam and increases its energy while maintaining a self-similar shape [22,23]. In this regime, the radiation pulse has a peak power that increases with the square of the distance z along the undulator and a longitudinal width that decreases with the inverse square root of z.…”

“…The electron beam was injected in the undulator system, consisting of six sections of 75 periods each, with u ¼ 2:8 cm [26]. Superradiance was induced, as in the configuration tested in [23], by seeding the FEL as a single-pass amplifier, with a short laser pulse of peak power comparable to the FEL saturation intensity. The undulators were all tuned at r ¼ seed and the Twiss parameter h i ' 1:5 m ensured the optimum beam matching.…”