In the last few decades, the development of lasers has followed two distinct directions. In one of these directions, scientists have pushed the limit to reach multi-petawatt (10¹⁵ W) short laser pulses on the basis of chirped-pulse amplification;1,2 in the other, efforts are under way to shorten the wavelength of the emitted radiation as much as possible.3,4 In his 2003 Nobel Prize speech, Ginzburg identified the gamma-ray laser as one of the challenges of the 21st century.5

One of the possible sources of high-frequency radiation is emission by electrons interacting with a strong optical laser pulse (nonlinear Compton scattering, NCS). For a single particle, the spectrum of this radiation is well known.6,7 Naively, the radiation of N particles should simply be N times stronger. However, if the motion of the particles is strongly correlated, i.e., if the initial conditions for spatial location8,9 and velocity10 are close (meaning that the density of the particles is high and the temperature is low), then the radiated energy can be strongly enhanced due to coherency8 (see Fig. 1).

Indeed, the energy of the emitted radiation in an external field at the classical limit is given by6$$\frac{\mathrm{d}{\mathcal{E}}_k}{\mathrm{d}\mathbf{k}}=-\frac{j^{\mu }(k)j_{\mu }^{\star }(k)}{4{\pi }^2},$$(1)where k is the radiation wave vector and j is the Fourier transform of the current 4-vector. For two emitting particles,$$|j^{\mu }{|}^2=\underset{\text{incoherent\hspace{0.17em}term}}{\underbrace{|j_1^{\mu }{|}^2+|j_2^{\mu }{|}^2}}+\underset{\text{interference\hspace{0.17em}term}}{\underbrace{2\mathsf{R}\mathsf{e}(j_1^{\mu }{j}_{\mu 2}^{\star })}},$$(2)and under certain conditions, the interference term can be as large as the incoherent term, increasing the radiated power by a factor of 2. These conditions for the particles’ initial displacement R were established in Ref. 8: R is inversely proportional to the emitted frequency ω, and it also depends on the particles’ initial energy and the intensity of the laser.

For N particles, the number of incoherent terms scales as N, while the number of interference terms scales as N². This means that the coherent enhancement of radiation for a macroscopic number of particles can be huge. Since R ∼ 1/ω, the lower frequencies are enhanced more strongly than the higher frequencies; i.e., the shape of the radiation spectrum (with respect to the single-particle spectrum) is substantially modified by the coherency (see Fig. 1). Note that since the average distance between the particles is defined by the density n, the spectrum is very sensitive to the density. This is revealed in two particular ways: first, the enhancement factor for each frequency is proportional to the density; second, the maximum frequency ω_coh, which is enhanced due to coherency, scales as ω_coh ∼ n^1/3.

It is worth noting that despite “higher” frequencies being unaffected by coherency, the threshold ω_coh of coherently enhanced “low” frequencies can be quite high—up to tens (and in some conditions even hundreds) of keV, depending on the laser intensity and the particles’ initial energy.8 In particular, for a 50-MeV electron beam with a density of 10²² cm⁻³ and a laser intensity of 10¹⁹ W/cm², the coherency threshold exceeds 1 keV (see Fig. 1), while a 10-keV coherency threshold is provided by an energy of 200 MeV and a solid density of 10²³ cm⁻³.8

Moreover, according to Ref. 8, the ratio of the energy of the coherently enhanced radiation to that of the incoherent radiation (for initially slow particles and a relativistically intense laser a₀ ≫ 1) is$$\frac{{\mathcal{E}}_{\text{coh}}}{{\mathcal{E}}_{\text{incoh}}}\sim \frac{1}{64{\pi }^3}\frac{n{\lambda }_{\text{L}}^3}{a_0^2},$$(3)where a₀ = eE_Lλ_L/2πmc² is the dimensionless laser amplitude (in which E_L and λ_L are the laser’s electric-field amplitude and wavelength, −e and m are the electron charge and mass, and c is the speed of light), meaning that the total energy of the coherent part of the spectrum can be much higher than that of the incoherent part. Indeed, for solid-density electrons and an optical laser pulse, the factor $n{\lambda }_{\text{L}}^3$ can be as high as 10¹¹, and for a₀ ∼ 10² (corresponding to an intensity I ∼ 10²² W/cm²), one obtains ${\mathcal{E}}_{\text{coh}}/{\mathcal{E}}_{\text{incoh}}\sim 10^4$.

Although the creation of a “graser” (gamma-ray laser) is not imminent, it is clear that possibilities do exist for generating coherent hard x-rays using collective effects in plasmas when interacting with a short and ultra-intense laser pulses in the infrared regime. Optimization of the interaction process allows the coherent and incoherent contributions of the radiation generated by nonlinear inverse Compton scattering to be influenced; partial control of the coherency is thus possible.

Nevertheless, a number of important theoretical questions still remain, and these need to be addressed in subsequent refined research. First, the presented approach is purely classical, and for an accurate investigation of the emission at high frequencies, one has to develop a quantum approach with accounting for coherency. Second, the effect of the temperature of the particle ensemble should be taken into consideration. Third, when coherency is taken into account, the particles can radiate so strongly [see Eq. (3) and the discussion below] that even for a mildly relativistic bunch and a moderately intense laser (I ≳ 10¹⁸ W/cm²) this will modify their trajectories in the laser field (the radiation reaction effect6), which will in turn affect the radiation emission. Finally, for a detailed description of laser–electron-bunch collision—which is necessary for planning and interpretation of experiments—numerical simulations are required. Particle-in-cell simulations are very powerful tools for investigation of the interaction of a plasma with a short intense laser pulse. Developing an implementation of coherent photon emission in such simulations is a challenging but necessary step toward employing coherent NCS for the creation of a hard x-ray source.

Acknowledgment. E.G.G. is grateful to M. Grech, A. Grassi, and A. Mironov from LULI (France) for fruitful discussions. The collaboration with LULI was supported by the Czech Academy of Sciences (Mobility Plus Project No. CNRS-23-12). A.M.F. was supported by the Russian Science Foundation (Grant No. 20-12-00077).