# Comments on the Physics of Microwave-Undulators
## 1 Introduction  
The electromagnetic field constrained in a waveguide can impress an oscillatory motion to an electron (or to an electron beam) moving through it. The action is therefore not dissimilar from the role played by the static field of a magnetic undulator, constituting the core of any Free Electron Laser (FEL) device. Accordingly, the high power of microwave fields passing through a waveguide could provide the essential tool for the realization of Wave Undulator (WU) based FELs. Although the concept is by no means new and traces back to the 1980s (see [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] for an albeit partial list), the development of high powerC/X-band sources (and not only) for high gradient accelerators [10] could provide the technologies necessary for an effective development of WU-FELs. The advantages offered by this solution are evident in terms of radiation polarization, effect of wake field mitigation on the beam itself (with the consequent possibility of operating with larger currents), and reduction of costs and physical size. The disadvantages are easily summarized by observing that in spite of the claimed perspective superiority with respect to the its magnetic counterpart and in spite of the fact that the relevant proposals are more than forty years old, no WU-FEL has been operated so far. The spontaneous radiation from an S-Band wave and electron beam with energy above 100 MeV has been experimentally studied in ref. [5]. This paper had many merits and revealed many interesting features, the first of which is that of putting in clear evidence that, by using the adequate correspondences, the magnetic and WU bremsstrahlung processes can be treated on the same ground. This aspect of the problem was initially studied in [6], and in ref. [11], a proposal has been put forward for the construction of a short wavelength FEL operating with an X-Band WU. In more recent times, the problem has been further studied and different types of WU have been proposed, along with various types of FEL architectures, including laser wave undulators as well (see [19] and reference therein). Even though we have not specified what the drawbacks of a WU-FEL are, it is evident that most of the problems arise from the large amount of microwave power necessary to feed the cavity to reach a reasonable strength parameter from the mechanical machining of the waveguide itself and from the power attenuation along the propagation. All these effects cause inhomogeneous broadening, which combine with the other well-known emission line degrading effects, thus inducing further potentially harmful contributions degrading the FEL gain and its performances. In this paper, we take advantages from the previously quoted studies to propose a comprehensive and accurate investigation of the microwave-electron interaction mechanisms to address a more appropriate direction to the strategies developed for the design of FELs operating with microwave undulators. In the past, the problem has been treated using a rather pragmatic point of view. Most of the treatment has been that of \"viewing\" the microwave field as static, thus associating to it an equivalent period, a peak magnetic field and, accordingly, an equivalent strength parameter. This procedure is in some sense the inverse of the Weiszacker-Williams approximation (WW-A) [20] exploited by [PERSON] in his seminal paper [21] to treat the emission process by relativistic electron in undulators as a scattering process of the undulator associated virtual photons [22; 23; 24]. The reverse point of view is, in principle, legitimate. It should be, however, properly framed within the context of an accurate understanding of the equations modeling the spectral details of the radiation emitted by relativistic electrons propagating in a microwave undulator. Before entering the core of the discussion developed in the forthcoming sections, we summarize the points we will carefully treat in the following.