Equivalent photons, which represent the field of a highly relativistic charged particle moving with a constant velocity, may closely resemble real photons, which forms the basis of the Weizsäcker-Williams approximation widely used in high-energy physics (for a review see, e.g., Ref. [16]). A simple and illustrative discussion of this approximation can be found in [17].

Unlike real (physical) photons in vacuum, equivalent (“virtual”) photons, which are used for an effective description of the electromagnetic field surrounding a moving charge, are off-shell, in addition to transverse polarization can contain longitudinal polarization, cannot “live” independently, and thus cannot be observed as independent particles.

The bunch density and volume are taken at an arbitrarily fixed moment of time.

In the rest frame of the beam, the atomic transitions are caused by inelastic scattering of an extreme relativistic atom on a diffraction grating composed of groups of “static” electrons.

In the standard seeding scheme, many harmonics of the seed laser are present in the electron density modulation.

The main effects influencing the bunch shape after leaving the undulator are the following. First, once the FEL operates near or at saturation, the electron bunch exits the undulator with substantial coherent energy modulation. A relevant scale in the undulator is the FEL gain length $l_g$; the corresponding scale in the drift is $l_g(1+K^2/2)$, since the longitudinal dispersion is reduced by the factor $1+K^2/2$. A reverse taper [27] can furthermore suppress the energy modulation while preserving strong microbunching. Second, debunching arises from the uncorrelated spread in longitudinal velocities, predominantly due to second-order coupling between longitudinal and transverse motion; the relevant parameter is given in [14]. Third, longitudinal space charge limits the drift length to the inverse plasma oscillation frequency, $c/{\omega }_p$. Our estimates, which take into account the above effects, showed that the bunch keeps its periodic structure over distances from several meters (for our FLASH example) to several tens of meters (for our European XFEL example).

The parameters $a_0$ and $l_0$ are related to the corresponding root-mean-square values of the beam by $\sqrt{⟨X^2⟩}=\sqrt{⟨Y^2⟩}=a_0/\sqrt{2}$ and $\sqrt{⟨Z^2⟩}=l_0/\sqrt{2}$.

At fixed values of $N_t$, $\gamma$, ${\lambda }_U$, and $\mathcal{K}$, the number of microbunches and the number of electrons in a single microbunch are proportional to $L$ and $1/L$, respectively. The intensity of photons, coherently created by a single microbunch, is proportional to the square of the number of the electrons in the microbunch (i.e., $\sim 1/L^2$). The coherent action of the microbunches increases the peak intensity of each of the CE lines by a factor proportional to the square of the number of the microbunches (i.e., $\sim L^2$), whereas the width of the CE lines is inversely proportional to the number of the microbunches (i.e., $\sim 1/L$). As a result, the integrated intensity of each CE line turns out to be proportional to $1/L$. This conclusion also follows from our numerical calculations.

The variation of the dependence on the beam radius from $\sim a_0^{-4}$ at $a_0\gg v\gamma /\omega$ to $\sim \ln (1/a_0)$ at $a_0\ll v\gamma /\omega$, where $\omega$ is the frequency of the equivalent photon, is similar both for periodically modulated electron beams and for beams consisting of just a single bunch. For the discussion of the dependence on the beam radius $a_0$ in the latter case, see Ref. [10].

The shape is mainly determined by the factor $\exp (-{\omega }_{\mathrm{ph}}^2{L}^2/v^2)$.

In our consideration we neglect electrodynamic environment but note that $b\sim 100a_0$ would already approach the radius of vacuum chamber that prevents from considering larger values and strongly modifies the low-frequency part of the spectrum.

Tunneling ionization is a highly nonperturbative process and, in order to calculate its cross section, we used the approach described in detail in [10], which employs tunneling rate expressions from [36] valid for ionization by a constant or slowly varying electric field.

In the FLASH case the tunneling is negligible because the electron bunch is longer and broader than that in the XFEL case: this reduces the field strength, having a dramatic effect on the tunneling rates due to their extreme high sensitivity to the field strength. Since the tunneling rate is also extremely sensitive to the typical interatomic field, an increase in the binding energy of atomic electrons also has a dramatic effect on the tunneling ionization: for instance, replacing hydrogen by helium reduces the tunneling cross section, shown in Fig. 4, by several orders of magnitude. However, even though one can expect that the low-frequency field will normally have a very modest (direct) influence on atomic ionization, effects of a concerted action of the high- and low-frequency fields may still remain, since even a relatively weak low-frequency field can be very efficient in mixing closely lying atomic states with opposite parities.

For an illustration of how the ratio between the ionization by the absorption of the first high-energy CE line and by the individual action of the bunch electrons varies with the change in the binding energy of the target electron we present here the corresponding cross sections (in $\mu {\mathrm{m}}^2$) in collisions with the bunch, whose parameters are given in Fig. 4 and assuming that $a_0=10\mathrm{\mu }\mathrm{m}$: ${\sigma }_{\mathrm{CE}}=2.5\times {10}^{-4}$ and ${\sigma }_{\mathrm{ind}}=4.5\times {10}^{-2}$ for $\mathrm{He}(1{\mathrm{s}}^2)$; ${\sigma }_{\mathrm{CE}}=7.5\times {10}^{-4}$ and ${\sigma }_{\mathrm{ind}}=6.8\times {10}^{-3}$ for ${\mathrm{He}}^{+}(1\mathrm{s})$; ${\sigma }_{\mathrm{CE}}=4.4\times {10}^{-3}$ and ${\sigma }_{\mathrm{ind}}=2.9\times {10}^{-3}$ for ${\mathrm{Li}}^{2+}(1\mathrm{s})$; ${\sigma }_{\mathrm{CE}}=1.43\times {10}^{-2}$ and ${\sigma }_{\mathrm{ind}}=1.63\times {10}^{-3}$ for ${\mathrm{Be}}^{3+}(1\mathrm{s})$; ${\sigma }_{\mathrm{CE}}=3.45\times {10}^{-2}$ and ${\sigma }_{\mathrm{ind}}=1.0\times {10}^{-3}$ for ${\mathrm{B}}^{4+}(1\mathrm{s})$; ${\sigma }_{\mathrm{CE}}=6.9\times {10}^{-2}$ and ${\sigma }_{\mathrm{ind}}=7.0\times {10}^{-4}$ for ${\mathrm{C}}^{5+}(1\mathrm{s})$; ${\sigma }_{\mathrm{CE}}=0.12$ and ${\sigma }_{\mathrm{ind}}=5.1\times {10}^{-4}$ for ${\mathrm{N}}^{6+}(1\mathrm{s})$; ${\sigma }_{\mathrm{CE}}=0.19$ and ${\sigma }_{\mathrm{ind}}=3.9\times {10}^{-4}$ for ${\mathrm{O}}^{7+}(1\mathrm{s})$. The number of the CE photons in the first line ($\hslash {\omega }_1\approx 1\mathrm{keV}$) is $\approx 3\times {10}^{10}$ and the line energy is $\approx 3\times {10}^{13}\mathrm{eV}$. We also note that for all the above targets the tunnel ionization is negligible.

In this case the tunneling ionization is very weak, but the atomic electrons can also be removed by collisions with individual electrons. Our rough estimates for the cross section for the removal of a $2p$ ($2s$) electron yield $\simeq 0.1{\mathrm{\mu m}}^2$ ($\simeq 0.015{\mathrm{\mu m}}^2$). For the cross section for the removal of an electron from the $K$ shell we obtain $\simeq 7.8\times {10}^{-4}{\mathrm{\mu m}}^2$, which is by 2 orders of magnitude smaller than the cross section for the removal of such an electron by the absorption of the CE photon from the first line ($\approx 1\mathrm{keV}$). We also note that the removal of a 2s electron from Ne does not trigger Auger process (it is not allowed energetically) and, therefore, is unlikely to lead to the production of ${\mathrm{Ne}}^{2+}$ ions.

Compared to the high-frequency CE lines, the field, which is generated by individual electrons, can also be regarded as an effectively low-frequency field. For a simple discussion of properties of such a field, expressed in terms of equivalent photons, see, e.g., Ref. [17].

Our results for the emission spectra, shown in Fig. 1, are obtained within the first-order perturbation theory in the interaction with the bunch field. This approximation does not take into account that the emitted electrons can be influenced by the field of the bunch (predominantly by its low-frequency part).

For instance, electron bunches with azimuthal symmetry generate equivalent photons that are polarized along the bunch radius.

The impact of the chicane on longitudinal dynamics can be optimized by operating the FEL at the onset of saturation and avoiding smearing effects associated with the uncorrelated energy spread.

As was stressed in the introduction, the field of an electron beam with a periodic space-time structure is tremendously influenced by a multilevel coherence. As a result, such beams affect atoms via a kind of “3-in-1” action: the synchronized interactions with (i) high-frequency CE photons and (ii) the low-frequency field of the beam, which both are coherently created by the beam electrons, are superimposed on (iii) the “background” interaction with the individual electrons. The first two very closely resemble the interactions with intense laser fields, whereas the third one proceeds in the standard regime of collisions with charged particles. By playing with the relative intensities of these channels one can, in particular, bridge the gap between the “standard” collisions and the interaction with intense laser fields.

The results for the cross sections do not depend on the choice of the origin: it can be an arbitrary point that is at rest in the laboratory frame.