FEL SASE and wave undulators

Abstract

We investigate the working conditions of Free Electron Lasers operating in the SASE regime with wave undulators. We provide general scaling criteria and corroborate them with appropriate numerical simulations.

Introduction

Free Electron Lasers (FEL) operating with undulators provided by electromagnetic waves were considered in the past [1], [2] and recently discussed again [3], [4] because of the development of laser technology and the availability of the electron beams with unprecedented large brightness [5].¹ High power lasers have undergone a spectacular evolution, the record intensity of $2\times {10}^{26}(\mathrm{W}/{\mathrm{m}}^2)$ has been obtained [6], and the upcoming petawatt lasers aim at intensities of the order of ${10}^{27}(\mathrm{W}/{\mathrm{m}}^2)$ [7]. In the near future, intensities of the order of ${10}^{28}-{10}^{30}(\mathrm{W}/{\mathrm{m}}^2)$ are foreseen at the Extreme-Light-Infrastructure (ELI) [8].

The advantages offered by such an equipped FEL are evident. Electron beam with modest energies could in principle be used to reach the hard X-ray region, as well as the construction of long undulators could be avoided. In contrast, high power lasers, challenging electron beam qualities and handlings, are necessary to provide a set of parameters, supporting the growth of a FEL SASE up to the saturation.

In this paper, we will not introduce new concepts and/or different experimental solutions, but we will derive a set of scaling formulae useful for a first design of a linearly polarized wave undulator based FEL SASE. We will study the conditions allowing such an operating regime and check its relevant validity using analytical and numerical analyses.

In an ordinary SASE FEL, electrons propagate in magnetic undulators characterized by a period λ_u, by an on axis field B₀ and by a strength parameter K. The fact that a FEL can operate with a wave undulator is not surprising. Pantell, Soncini and Puthoff pointed out the possibility of obtaining a laser-like emission by means of the stimulated electron scattering of a CO₂ laser (see Ref. [9a]), while Madey described the electron–undulator interaction using the Weiszacker-Williams method of virtual quanta (see Ref. [9b]). Accordingly the radiation emission by the electrons is treated as a Compton backscattering of the pseudo photons of the undulator, which is viewed as an electromagnetic wave with wavelength λ⁎ twice the period λ_u of the undulator itself.

Within such a point of view the selection of the wavelength λ_s (the resonance condition) of the emitted radiation is, accordingly, the result of a double Doppler shift of the undulator “photons” wavelength, according to the prescriptions:

$${\lambda }_s=\frac{1-\beta }{1+\beta }{\lambda }^{⁎}=2\frac{1-\beta }{1+\beta }{\lambda }_u,$$

where γ is the Lorentz factor and β the normalized velocity of the electron beam. Including in the analysis the transverse motion induced by the interaction and associated with the number of pseudo-photon density of the undulator field [10], one obtains

$${\lambda }_s\cong \frac{{\lambda }^{⁎}}{4{\gamma }^2}\left(1+\frac{K^2}{2}\right),$$

where

$$K=\frac{e{B}_0{\lambda }_u}{2\pi m_0{c}^2}$$

The strength parameter K plays a role of paramount importance in the case of laser undulators. It can, indeed, be understood as the electron mass intensity dependence shift, discussed by Sengupta [11] and later developed by Brown and Kibble [12] in their treatment of the Compton scattering by an intense electromagnetic wave.²

The dynamics of FEL operation with magnetic or wave undulators is essentially the same, if we limit ourselves to modest electron energies and not too short laser wavelengths, in order to avoid quantum corrections. The scaling laws reported in Ref. [13] will, therefore, be used to derive practical formulae yielding the conditions to be satisfied by the wave laser intensity and by the electron beam qualities to guarantee a reliable operation.

The paper consists of four sections. Section 2 is devoted to general considerations on the feasibility criteria of the device and on some scaling predictions on output power and saturation length. The relevant numerical test is reported in Section 3, and, finally, Section 4 contains critical remarks on the validity limits of the treatment and on the experimental feasibility of a wave undulator FEL.