Within the framework of quantum mechanics, a unique particle wave packet exists in the form of the Airy function. Its counterintuitive properties are revealed as it propagates in time or space: the quantum probability wave packet preserves its shape despite dispersion or diffraction and propagates along a parabolic caustic trajectory, even though no force is applied. This does not contradict Newton's laws of motion, because the wave packet centroid propagates along a straight line. Nearly 30 years later, this wave packet, known as an accelerating Airy beam, was realized in the optical domain; later it was generalized to an orthogonal and complete family of beams that propagate along parabolic trajectories, as well as to beams that propagate along arbitrary convex trajectories. Here we report the experimental generation and observation of the Airy beams of free electrons. These electron Airy beams were generated by diffraction of electrons through a nanoscale hologram, which imprinted on the electrons' wavefunction a cubic phase modulation in the transverse plane. The highest-intensity lobes of the generated beams indeed followed parabolic trajectories. We directly observed a non-spreading electron wavefunction that self-heals, restoring its original shape after passing an obstacle. This holographic generation of electron Airy beams opens up new avenues for steering electronic wave packets like their photonic counterparts, because the wave packets can be imprinted with arbitrary shapes or trajectories.

This article presents a unified formulation and review of an extensive class of radiation effects and devices based on free or quasifree electrons. The effects and devices reviewed include slow-wave radiators [such as Cerenkov, Smith-Purcell, and TWT (traveling-wave tube) effects and devices], periodic bremsstrahlung radiators [such as undulator radiation, magnetic bremsstrahlung FEL's (free-electron lasers), and coherent bremsstrahlung in the crystal lattice], and transverse-binding radiators [such as the CRM (cyclotron resonance maser) and channeling radiation]. Starting from a general quantum-electrodynamic model, both quantum and classical effects and operating regimes of these radiation devices are described. The article provides a unified physical description of the interaction kinematics, and presents equations for the characterization of spontaneous and stimulated radiative emission in these various effects and devices. Universal relations between the spontaneous and stimulated emission parameters are revealed and shown to be related (in the quantum limit) to Einstein relations for atomic radiators and (in the classical limit) to the relations derived by Madey for magnetic bremsstrahlung FEL for on-axis radiative emission. Examples for the application of the formulation are given, estimating the feasibility of channeling radiation x-ray laser and optical regime Smith-Purcell FEL, and deriving the gain equations of magnetic bremsstrahlung FEL and CRM for arbitrary electron propagation direction, structure (wiggler) axis, and radiative emission angle.

A formulation for the characterization of superradiant and stimulated-superradiant radiative emission from bunched electron beams is presented. The radiation is characterized in terms of power and spectral power per radiation mode, which provide a measure of the useful spatially coherent radiation power and spectral power emitted by a radiation source. When the bunched electron beam emits superradiantly, these parameters scale like the square of the number of electrons, orders of magnitude more than spontaneous emission. The formulation applies to emission from single bunches, a finite number of bunches in a macropulse, or periodic bunching. It can be employed on any kind of e-beam radiation scheme. Specific analytic expressions are derived for coherent synchrotron radiation and prebunched free-electron laser, providing a basis for comparing and understanding their connection.

In the foundation of quantum mechanics, the spatial dimensions of electron wavepacket are understood only in terms of an expectation valuethe probability distribution of the particle location. One can still inquire how the quantum electron wavepacket size affects a physical process. Here we address the fundamental physics problem of particle-wave duality and the measurability of a free electron quantum wavepacket. Our analysis of stimulated radiative interaction of an electron wavepacket, accompanied by numerical computations, reveals two limits. In the quantum regime of long wavepacket size relative to radiation wavelength, one obtains only quantum-recoil multiphoton sidebands in the electron energy spectrum. In the opposite regime, the wavepacket interaction approaches the limit of classical point-particle acceleration. The wavepacket features can be revealed in experiments carried out in the intermediate regime of wavepacket size commensurate with the radiation wavelength.      , where   0ˆz Ee z iq z z E e  represents the dominant slow component of the radiation wave, and transverse field components, as well astransverse variation of the field are neglected. We examplify our modeling here for a case of Smith-Purcell radiation (see Fig. 1), for which the radiation wave is a Floquent mode: 4   zm iq z m m ze   EEwith 0 2 zm z G q q m   , G  is the grating period, 0 Θ, / z q qcos q c   and Θ is the incidence angle of the radiation wave relative to the axial interaction dimension. The radiation wave number z zm qq  represents one of the space harmonics m that satisfies synchronizm condition with the electron [6]: 0 zm vq   .We note that the analysis would be similar for the Cerenkov interaction scheme with   Θ z q n cos   , and () n  the index of refraction of the medium. Furthermore, the analysis can be extended to the case of FEL and other interaction schemes [13,14].The solution of Schrodinger equation to zero order (i.e. free-space propagation) is well known. Assuming that the initial wavepacket, which is emitted at some point