Smith–Purcell radiation of a vortex electron

Analogously to optics, there are two main models of the vortex particles—the Bessel beams and the paraxial LG packets [1]. There is also a non-paraxial generalization of the latter, called the generalized LG packet [18],

$$\begin{align}\hfill & {\psi }_{\ell ,n}\left(\boldsymbol{r},t\right)=\sqrt{\frac{n!}{\left(n+\vert \ell \vert \right)!}}\frac{{i}^{2n+\ell }}{{\pi }^{3/4}}\frac{{\rho }^{\vert \ell \vert }{\ell }^{\frac{2\vert \ell \vert +3}{4}}}{{\left(\bar{\rho }\left(t\right)\right)}^{\vert \ell \vert +3/2}}\enspace {L}_{n}^{\vert \ell \vert }\left(\frac{\ell {\rho }^{2}}{{\left(\bar{\rho }\left(t\right)\right)}^{2}}\right)\enspace \mathrm{exp}\left\{-it\frac{{\langle p\rangle }^{2}}{2m}+i\langle p\rangle z+i\ell {\phi }_{r}\right.\hfill \\ \hfill & \left.\qquad -i\left(2n+\vert \ell \vert +\frac{3}{2}\right)\mathrm{arctan}\left(\frac{t}{{t}_{d}}\right)-\frac{\ell \left(\left({t}_{d}-it\right)\right)}{2{t}_{d}{\left(\bar{\rho }\left(t\right)\right)}^{2}}\enspace \left({\rho }^{2}+{\left(z-\langle u\rangle t\right)}^{2}\right)\right\},\hfill \\ \hfill & \qquad {\times}\int {\mathrm{d}}^{3}r\enspace \vert {\psi }_{\ell ,n}\left(\boldsymbol{r},t\right){\vert }^{2}=1,\quad \rho =\sqrt{{x}^{2}+{y}^{2}},\hfill \end{align} \tag{ 2 }$$

which represents an exact non-stationary solution to the Schrödinger equation and whose centroid propagates along the z axis. The paraxial LG packets and the Bessel beams represent two limiting cases of this model [23]. Below we consider this packet with n = 0 only. Note that the factor 3/2 in the Gouy phase in (2) comes about because the packet is localized in a three-dimensional space (cf equation (61) in [23]).

In what follows, we employ the mean radius of the wave packet ${\bar{\rho }}_{0}$ instead of the beam waist σ_⊥,

$$\begin{equation*}{\sigma }_{\perp }={\bar{\rho }}_{0}/\sqrt{\vert \ell \vert },\hspace{25.0pt}\bar{\rho }\left(t\right)={\bar{\rho }}_{0}\sqrt{1+{t}^{2}/{t}_{d}^{2}}.\end{equation*}$$

As explained in [24], depending on the experimental conditions one could either fix the beam waist, and so the mean radius ${\bar{\rho }}_{0}$ scales as $\sqrt{\vert \ell \vert }$, or fix the radius itself. In this paper, we follow the latter approach and treat ${\bar{\rho }}_{0}$ as an OAM-independent value. An approach with the fixed beam waist σ_⊥ can easily be recovered when substituting ${\rho }_{0}\to {\sigma }_{\perp }\enspace \sqrt{\vert \ell \vert }$.

Although non-paraxial effects are nearly always too weak to play any noticeable role, the difference between this non-paraxial LG packet and the paraxial one becomes crucial for a moderately relativistic particle with β ≲ 0.9. In this regime, which is the most important one for the current study, it is only equation (2) that yields correct predictions for observables and is compatible with the general CPT-symmetry [18]. As we argue below, the non-paraxial effects are additionally enhanced when the packet's spreading becomes noticeable and the OAM is large.

The packet spreads and its transverse area is doubled during a diffraction time t_d ,

$$\begin{equation}{t}_{d}=\frac{m{\bar{\rho }}_{0}^{2}}{\vert \ell \vert }=\frac{{t}_{c}}{\vert \ell \vert }\enspace {\left(\frac{{\bar{\rho }}_{0}}{{\lambda }_{c}}\right)}^{2}\gg {t}_{c},\end{equation} \tag{ 3 }$$

which is large compared to the Compton time scale t_c = λ_c /c ≈ 1.3 × 10⁻²¹ s, λ_c ≈ 3.9 × 10⁻¹¹ cm. When the LG packet moves nearby the grating, the finite spreading time (3) puts an upper limit on the possible impact parameter h, on the initial mean radius of the packet and on the grating length. Indeed, both the corresponding solution of the Schrödinger equation and the multipole expansion have a sense only for as long as $\bar{\rho }\left(t\right){< }h$ or, alternatively,

$$\begin{equation}\frac{t}{{t}_{d}}{< }\sqrt{\frac{{h}^{2}}{{\bar{\rho }}_{0}^{2}}-1}.\end{equation} \tag{ 4 }$$

Let ${t}_{\mathrm{max}}={t}_{d}\sqrt{\frac{{h}^{2}}{{\bar{\rho }}_{0}^{2}}-1}$ be the time during which the packet spreads to the extent that it touches the grating ⁴ ; then the corresponding number of strips N_max is

$$\begin{equation}{N}_{\mathrm{max}}=\beta {t}_{\mathrm{max}}/d=\frac{\beta }{\vert \ell \vert }\enspace \frac{{\bar{\rho }}_{0}}{{\lambda }_{c}}\enspace \frac{h}{d}\enspace \sqrt{1-\frac{{\bar{\rho }}_{0}^{2}}{{h}^{2}}}.\end{equation} \tag{ 5 }$$

The geometry implies that ${\bar{\rho }}_{0}{< }h=\bar{\rho }\left({t}_{\mathrm{max}}\right)$ or ${\bar{\rho }}_{0}\ll h$ for the long grating. In practice, only small diffraction orders of the radiation can be considered, so that d ∼ βλ for the emission angles Θ ∼ 90°. So an upper limit for the number of strips in this case is

$$\begin{equation}{N}_{\mathrm{max}}\lesssim \frac{1}{\vert \ell \vert }\enspace \frac{{\rho }_{0}}{{\lambda }_{c}}\enspace \frac{h}{\lambda }.\end{equation} \tag{ 6 }$$

If this condition is violated, the multipole expansion is no longer applicable. A rough estimate of the maximal number of strips for $h\approx {h}_{\text{eff}}\sim 0.1\lambda ,\beta \approx 0.5,{\bar{\rho }}_{0}\sim 1$ nm yields

$$\begin{equation}{N}_{\mathrm{max}}\lesssim \frac{1{0}^{3}}{\vert \ell \vert }.\end{equation} \tag{ 7 }$$

So if N_max ≫ 1, then |ℓ| < 10³.

The LG electron packet carries, in addition to the charge, higher multipole moments [25, 26]. In particular, at the distances larger than the mean radius of the vortex packet,

$$\begin{equation}r\gtrsim \bar{\rho }\left(t\right),\end{equation} \tag{ 8 }$$

it is sufficient to keep a magnetic dipole moment and an electric quadrupole moment [18, 23],

$$\begin{equation}\boldsymbol{\mu }=\hat{\boldsymbol{z}}\enspace \frac{\ell }{2m}\left(1-\frac{1}{2}\enspace {\ell }^{2}\enspace \frac{{\lambda }_{c}^{2}}{{\bar{\rho }}_{0}^{2}}\right),\hspace{25.0pt}\enspace {Q}_{\alpha \beta }\left(t\right)={\left(\bar{\rho }\left(t\right)\right)}^{2}\enspace \mathrm{diag}\left\{1/2,1/2,-1\right\}.\end{equation} \tag{ 9 }$$

The magnetic moment includes a non-paraxial correction according to equation (45) in [23], which is written for the case |ℓ| ≫ 1. The field of the electron's quadrupole moment originates from a non-point source as the quadrupole has a finite width, which is just equal to an rms-radius of the packet.

Note that although an LG packet with ℓ = 0 has a vanishing quadrupole moment, an OAM-less packet with a non-vanishing quadrupole momentum can be easily constructed by making this packet highly asymmetric in shape. Thus our conclusions below can also be applied to an arbitrary non-symmetric wave packet with a non-vanishing quadrupole moment.

When the spreading is essential—at t ≳ t_d —the inequality (8) can be violated and the multipole expansion cannot be used at all. Thus, the conventional (paraxial) regime of emission takes place only when the spreading is moderate, t ≲ t_d . Remarkably, the non-paraxial regime of emission favors moderately large values of the OAM, in contrast to the enhancement of the magnetic moment contribution for which the OAM should be as large as possible. This is because the quadrupole moment has a finite radius and so the radiation is generated as if the charge were continuously distributed along all the coherence length and not confined to a point within this length [14].

Single-electron regime with a freely propagating packet is realized for low electron currents—below the so-called start current, which is typically about 1 mA [27]. The charge, the magnetic dipole and the electric quadrupole moments (9) induce surface currents on the grating. These currents, in their turn, generate electric and magnetic fields E _e , E _μ , E _Q , etc. The total radiation intensity dW includes the multipole radiation intensities as well their mutual interference, which serve as small corrections to the classical radiation from the point charge dW_ee :

$$\begin{equation}\frac{\mathrm{d}W}{\mathrm{d}\omega \enspace \mathrm{d}{\Omega}}\equiv \mathrm{d}W=\mathrm{d}{W}_{ee}+\mathrm{d}{W}_{e\mu }+\left(\mathrm{d}{W}_{eQ}+\mathrm{d}{W}_{\mu \mu }\right)+\left(\mathrm{d}{W}_{\mu Q}+\mathrm{d}{W}_{eO}\right)+\left(\mathrm{d}{W}_{QQ}+\mathrm{d}{W}_{e\mathrm{16}p}+\mathrm{d}{W}_{\mu O}\right)+\cdots \end{equation} \tag{ 10 }$$

In this paper, we adhere to such a perturbative regime and formally order perturbative corrections following the order of the multipole expansion. The leading order (LO) correction dW_eμ is given by the charge-magnetic-moment radiation. The next-to-leading (NLO) order corrections include the charge-electric-quadrupole radiation dW_eQ and the radiation of the magnetic moment dW_μμ . The next-to-next-to-leading (NNLO) order corrections already include the interference term between the magnetic momentum and electric-quadrupole radiation dW_μQ and the higher multipole term with the charge-octupole radiation dW_eO . The quadrupole–quadrupole radiation dW_QQ appears with interference terms from higher multipoles (octupole dW_μO and 16-pole dW_e16p ).

Moreover, in the overwhelming majority of practical cases it is sufficient to calculate the charge contribution and the following interference terms

$$\begin{equation}\mathrm{d}W=\mathrm{d}{W}_{ee}+\mathrm{d}{W}_{e\mu }+\mathrm{d}{W}_{eQ}\end{equation} \tag{ 11 }$$

only, while dW_μμ , dW_μQ , and the higher-order corrections can be safely neglected. We emphasize that it does not mean that there are simple inequalities like dW_ee ≫ dW_eμ ≫ dW_eQ ≫ dW_μQ ... for all the angles and frequencies. For instance, in the plane perpendicular to the grating, Φ = π/2, the term dW_eμ vanishes while dW_eQ does not (see below). This makes the region of angles Φ ≈ π/2 preferable for detection of the non-paraxial quadrupole effects.

An approach in which the particle trajectory is given holds valid when the quantum recoil η_q (ratio between the energy of the emitted photon and the electron's kinetic energy ɛ) is small compared to the interference corrections dW_eμ and dW_eQ ,

$$\begin{equation}{\eta }_{q}{:=}\frac{\omega }{\varepsilon }\ll \frac{\mathrm{d}{W}_{e\mu }}{\mathrm{d}{W}_{ee}},\quad \frac{\omega }{\varepsilon }\ll \frac{\mathrm{d}{W}_{eQ}}{\mathrm{d}{W}_{ee}},\end{equation} \tag{ 12 }$$

and the energy losses stay small compared to the electron's energy. We emphasize that the multipole contributions in equation (10) also have a quantum origin as they are due to non-Gaussianity of the wave packet or, in other words, due to its non-constant phase. So the series (10) is not quasi-classical. On a more fundamental level, there are two types of quantum corrections to the classical radiation of charge [28–32]:

• Those due to recoil,
• Those due to finite coherence length of the emitting particle.

While the quasi-classical methods like an operator method [32] and the eikonal method [31] neglect the latter effects from the very beginning and take into account the recoil only, here we demonstrate that there is an opposite non-paraxial regime of emission.

Let us study dimensionless parameters that define multipole corrections to the classical emission of a charge. The magnetic moment contribution (see equations (68) and (69)) is proportional to the following ratio:

$$\begin{equation}\frac{\mathrm{d}{W}_{e\mu }}{\mathrm{d}{W}_{ee}}\sim {\eta }_{\mu }{:=}\ell \frac{{\lambda }_{c}}{\lambda },\end{equation} \tag{ 13 }$$

which is of the order of ℓ ⋅ 10⁻⁷ for λ ∼ 1 μm, and ℓ ⋅ 10⁻¹⁰ for λ ∼ 1 mm. It is well known that a spin-induced magnetic moment contribution—the so-called 'spin light' [33]—and the recoil effects are of the same order of magnitude; thus we neglect both of them in our approach. However, for the vortex electron the magnetic moment contribution (13) is ℓ times enhanced, which legitimates the calculations of dW_eμ via the multipole expansion [2]. Indeed, for the electrons with β ≈ 0.4–0.8 and the kinetic energy of ɛ_c ∼ 50–300 keV, the small parameter governing the quantum recoil is

$$\begin{equation}{\eta }_{q}=\frac{\omega }{\varepsilon }\sim \frac{1}{m\lambda }\equiv \frac{{\lambda }_{c}}{\lambda },\end{equation} \tag{ 14 }$$

and η_q ≪ η_μ yields

$$\begin{equation*}\vert \ell \vert \gg 1,\end{equation*}$$

while the condition ${\eta }_{\mu }^{2}\ll {\eta }_{q}$ puts an upper limit on the OAM value,

$$\begin{equation}\vert \ell \vert \lesssim {\ell }_{\mathrm{max}}{:=}\sqrt{\frac{\lambda }{{\lambda }_{c}}}\sim {\eta }_{q}^{-1/2}\sim 1{0}^{3}-1{0}^{5},\quad \lambda \sim 1\enspace \mu \text{m}-1\enspace \text{mm},\end{equation} \tag{ 15 }$$

and so the contribution dW_μμ stays small.

As shown in section 3.1, one can distinguish three different corrections from the charge-quadrupole interference: $\mathrm{d}{W}_{e{Q}_{0}}$, $\mathrm{d}{W}_{e{Q}_{1}}$, and $\mathrm{d}{W}_{e{Q}_{2}}$. Their relative contributions are:

$$\begin{equation}\frac{\mathrm{d}{W}_{e{Q}_{0}}}{\mathrm{d}{W}_{ee}}\sim {\eta }_{{Q}_{0}}{:=}\frac{{\bar{\rho }}_{0}^{2}}{{h}_{\text{eff}}^{2}}-\text{quasi}-\text{classical}\;\text{quadrupole}\;\text{contribution},\end{equation} \tag{ 16 }$$

$$\begin{equation}\frac{\mathrm{d}{W}_{e{Q}_{1}}}{\mathrm{d}{W}_{ee}}\sim {\eta }_{{Q}_{1}}{:=}{\ell }^{2}\frac{{\lambda }_{c}^{2}}{{\bar{\rho }}_{0}^{2}}-\text{ordinary}\;\text{non}-\text{paraxial}\;\text{contribution}\;\left[\text{23}\right],\end{equation} \tag{ 17 }$$

$$\begin{equation}\frac{\mathrm{d}{W}_{e{Q}_{2}}}{\mathrm{d}{W}_{ee}}\sim {\eta }_{{Q}_{2}}{:=}{N}^{2}\enspace {\ell }^{2}\frac{{\lambda }_{c}^{2}}{{\bar{\rho }}_{0}^{2}}-\text{dynamically}\;\text{enhanced}\;\text{non}-\text{paraxial}\;\text{contribution}\;\left[\text{18}\right],\end{equation} \tag{ 18 }$$

where an effective impact parameter of S–P radiation naturally appears

$$\begin{equation}\enspace {h}_{\text{eff}}=\frac{\beta \gamma \lambda }{2\pi }=\frac{\beta \gamma }{\omega }\sim 0.1\enspace \lambda \enspace \quad \text{for}\enspace \beta \approx 0.4-0.8.\end{equation} \tag{ 19 }$$

The non-paraxial correction to the magnetic moment in (9) yields a correction to η_μ of the order of ${\eta }_{\mu }\enspace {\eta }_{{Q}_{1}}$, which can be safely neglected for our purposes.

As seen from (17), the non-paraxial regime does not necessarily imply a tight focusing, ${\bar{\rho }}_{0}\gtrsim {\lambda }_{c}$, but it can also be realized when the OAM is large, ℓ ≫ 1, whereas the focusing stays moderate, ${\bar{\rho }}_{0}\gg {\lambda }_{c}$. As we fix ${\bar{\rho }}_{0}$ and not the beam waist, the parameter (17) scales as ℓ², which for the electrons with ${\bar{\rho }}_{0}\sim 10$ nm and ℓ ∼ 10³ yields

$$\begin{equation*}{\ell }^{2}{\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2}\sim 1{0}^{-3},\end{equation*}$$

while it is 10⁻⁵ for ${\bar{\rho }}_{0}\sim 100$ nm.

Importantly, these non-paraxial (quadrupole) effects are dynamically enhanced when the packet spreading is essential on the radiation formation length, which for S–P radiation is defined by the whole length of the grating. Spreading of the packet with the time and distance ⟨z⟩ = tβ leads to growth of the quadrupole moment and a corresponding small dimensionless parameter ${\eta }_{{Q}_{2}}$ is

$$\begin{equation*}{t}^{2}/{t}_{d}^{2}\to {N}^{2}\end{equation*}$$

times larger than (17):

$$\begin{equation}{\eta }_{{Q}_{2}}{:=}{N}^{2}\enspace {\eta }_{{Q}_{1}}={N}^{2}\enspace {\ell }^{2}{\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2}.\end{equation} \tag{ 20 }$$

Thus the large number of strips N ≫ 1 can lead to the non-paraxial regime of emission with

$$\begin{equation}{\eta }_{{Q}_{1}}\ll 1,\hspace{25.0pt}{\eta }_{{Q}_{2}}\lesssim 1,\end{equation} \tag{ 21 }$$

when the quadrupole contribution becomes noticeable. Somewhat contrary to intuition, these non-paraxial effects get stronger when the packet itself gets wider, see (78).

In the OAM-less case, n = 0, ℓ = 0, the packet (2) turns into the ordinary, spherically symmetric Gaussian packet, which has a vanishing quadrupole and higher moments. Therefore, its spreading does not lead to such a non-linear enhancement and the S–P radiation from this packet in the wave zone coincides with that from a point charge (see also [14]). Note that if we fix the beam waist instead and, therefore, ${\bar{\rho }}_{0}\propto \sqrt{\vert \ell \vert }$, then

$$\begin{equation}{\eta }_{{Q}_{0}}=\mathcal{O}\left(\ell \right),\hspace{25.0pt}{\eta }_{{Q}_{1}}=\mathcal{O}\left(\ell \right),\hspace{25.0pt}{\eta }_{{Q}_{2}}=\mathcal{O}\left(\ell {N}^{2}\right).\end{equation} \tag{ 22 }$$

The dimensionless parameters from the NNL-order corrections dW_μμ , $\mathrm{d}{W}_{\mu {Q}_{j}}$, $\mathrm{d}{W}_{{Q}_{j}{Q}_{j}}$, dW_eO are just products of the leading and NL-order parameters,

$$\begin{equation}{\eta }_{\mu \mu }={\eta }_{\mu }^{2},\hspace{25.0pt}{\eta }_{\mu {Q}_{j}}={\eta }_{\mu }{\eta }_{{Q}_{j}},\hspace{25.0pt}{\eta }_{{Q}_{i}{Q}_{j}}={\eta }_{{Q}_{i}}{\eta }_{{Q}_{j}},\quad i=0,1,2,\enspace j=0,1,2.\end{equation} \tag{ 23 }$$

The following inequalities are in order

$$\begin{align}\hfill & {\eta }_{q}\ll {\eta }_{\mu },\hspace{25.0pt}{\eta }_{q}\ll {\eta }_{{Q}_{j}},\hspace{25.0pt}{\eta }_{\mu \mu }\lesssim {\eta }_{q},\hspace{25.0pt}{\eta }_{\mu {Q}_{j}}\lesssim {\eta }_{q},\hspace{25.0pt}{\eta }_{{Q}_{i}{Q}_{j}}\lesssim {\eta }_{q},\hfill \\ \hfill & i=0,1,2,\enspace j=0,1,2,\hfill \end{align} \tag{ 24 }$$

For the same beam energies, we have the following estimate for the first quadrupole parameter:

$$\begin{equation}{h}_{\text{eff}}\sim 0.1\enspace \lambda ,\hspace{25.0pt}{\eta }_{{Q}_{0}}\sim 1{0}^{2}\enspace \frac{{\bar{\rho }}_{0}^{2}}{{\lambda }^{2}}.\end{equation} \tag{ 25 }$$

According to (24) the inequalities ${\eta }_{{Q}_{0}}^{2}\lesssim {\eta }_{q}\ll {\eta }_{{Q}_{0}}$ restrict the initial rms-radius of the packet as follows:

$$\begin{equation}{\ell }_{\mathrm{max}}^{-1}\ll \frac{{\bar{\rho }}_{0}}{{h}_{\text{eff}}}\lesssim {\ell }_{\mathrm{max}}^{-1/2},\end{equation} \tag{ 26 }$$

which yields

$$\begin{align}\hfill & \lambda \sim 1\enspace \mu \text{m}:\enspace 0.1\enspace \text{nm}\ll {\bar{\rho }}_{0}\lesssim 3\enspace \text{nm},\hfill \\ \hfill & \lambda \sim 1\enspace \text{mm}:\enspace 1\enspace \text{nm}\ll {\bar{\rho }}_{0}\lesssim 300\enspace \text{nm}.\hfill \end{align} \tag{ 27 }$$

The packet radius should be smaller than the wavelength of the emitted radiation, which is just a condition of the multipole expansion in the wave zone.

The inequality ${\eta }_{{Q}_{1}}^{2}\lesssim {\eta }_{q}\ll {\eta }_{{Q}_{1}}$ defines either a lower bound on ℓ or an upper bound for the rms-radius:

$$\begin{equation}\vert \ell \vert \enspace {\lambda }_{c}\enspace {\ell }_{\mathrm{max}}^{1/2}\lesssim {\bar{\rho }}_{0}\ll \vert \ell \vert \enspace {\lambda }_{c}\enspace {\ell }_{\mathrm{max}},\end{equation} \tag{ 28 }$$

which yields

$$\begin{align}\hfill & 0.1-1\enspace \text{nm}\lesssim {\bar{\rho }}_{0}\ll 3\enspace \text{nm}-30\enspace \text{nm},\hspace{25.0pt}\ell \sim 10-100{< }{\ell }_{\mathrm{max}}\enspace \sim 1{0}^{3},\hspace{25.0pt}\lambda \sim 1\enspace \mu \text{m}\enspace \hfill \\ \hfill & 1-100\enspace \text{nm}\lesssim {\bar{\rho }}_{0}\ll 0.3-300\enspace \mu \text{m},\hspace{25.0pt}\ell \sim 10-1{0}^{4}{< }{\ell }_{\mathrm{max}}\enspace \sim 1{0}^{5},\hspace{25.0pt}\lambda \sim 1\enspace \text{mm}.\hfill \end{align} \tag{ 29 }$$

This is compatible with (27) provided that the OAM is at least ℓ ∼ 100.

Finally, the restrictions for the number of strips N can be derived from the inequality ${\eta }_{{Q}_{2}}^{2}\lesssim {\eta }_{q}\ll {\eta }_{{Q}_{2}}$:

$$\begin{equation}\frac{{\bar{\rho }}_{0}}{\vert \ell \vert {\ell }_{\mathrm{max}}{\lambda }_{c}}\ll N\lesssim {N}_{\mathrm{max}}{:=}\frac{{\bar{\rho }}_{0}}{\vert \ell \vert {\ell }_{\mathrm{max}}^{1/2}{\lambda }_{c}},\end{equation} \tag{ 30 }$$

where the ratio ${\bar{\rho }}_{0}/\left(\vert \ell \vert {\ell }_{\mathrm{max}}{\lambda }_{c}\right)$ itself must be less than unity according to (28). So, the smallest value of N can well be 1.

Let us estimate the largest possible number of strips for which our conditions hold. For an optical or infrared photon, λ ∼ 1 μm, ℓ ∼ 10², and ${\bar{\rho }}_{0}\sim 1\enspace \text{nm}-3\enspace \text{nm}$ (according to above findings), we get

$$\begin{equation*}{N}_{\mathrm{max}}\sim 3.\end{equation*}$$

So the grating must be really short. For a THz photon with λ ∼ 1 mm, ℓ ∼ 10², and ${\bar{\rho }}_{0}\sim 100\enspace \text{nm}-300\enspace \text{nm}$ we have

$$\begin{equation*}{N}_{\mathrm{max}}\sim 30,\end{equation*}$$

or the same number of N_max ∼ 3 for ℓ ∼ 10³. These inequalities specify the rough estimate (7). For S–P radiation, the large number of strips provides a narrow emission line, so the optimal OAM value is therefore

$$\begin{equation}\ell \sim 1{0}^{2}-1{0}^{3},\end{equation} \tag{ 31 }$$

and the optimal grating period, which defines the radiation wavelength as d ∼ λ, is

$$\begin{equation}d\sim 10-1000\enspace \mu \text{m}.\end{equation} \tag{ 32 }$$

For the largest wavelength, the maximal grating length for which the higher-multipole corrections can be neglected and the radiation losses stay small is of the order of 3 cm.

Importantly, the maximal grating length N_max d is much larger than the Rayleigh length z_R of the packet,

$$\begin{equation}{z}_{R}=\beta {t}_{d}=\beta \frac{{\lambda }_{c}}{\vert \ell \vert }\enspace {\left(\frac{{\bar{\rho }}_{0}}{{\lambda }_{c}}\right)}^{2},\end{equation} \tag{ 33 }$$

which is of the order of 0.1 μm for λ ∼ 1 μm and the same parameters as above, or z_R ∼ 1 mm for λ ∼ 1 mm and ℓ ∼ 10².

Summarizing, one can choose two baseline parameter sets:

• (IR): λ ∼ 1 μm, ${\bar{\rho }}_{0}=0.5-3\enspace \text{nm}$, ℓ ∼ 100, N ≲ 10,
• (THz): λ ∼ 1 mm, ${\bar{\rho }}_{0}=10-300\enspace \text{nm}$, ℓ ∼ 10² − 10³, N ≲ 100.

As has been already noted, in practice the corresponding inequalities and the subsequent requirements can often be relaxed, as the ratios like dW_eμ /dW_ee are generally functions of angles and frequency. For instance, the requirement η_q ≪ η_μ does not have a sense in a vicinity of Φ = π/2 as dW_eμ vanishes at this angle. Finally, note that typical widths of the electron packets after the emission at a cathode vary from several Angstrom to a few nm, depending on the cathode [34, 35], which meets our requirements.

Following the generalized surface current model developed in reference [15] we express the current density induced by the incident electromagnetic field of the electron on the surface of an ideally conducting grating as a vector product of E , the normal to the surface n and the unit vector to a distant point

$$\begin{equation*}{\boldsymbol{e}}_{0}=\frac{{\boldsymbol{r}}_{0}}{\vert {\boldsymbol{r}}_{0}\vert }=\left(\mathrm{sin}\enspace {\Theta}\enspace \mathrm{cos}\enspace {\Phi},\mathrm{sin}\enspace {\Theta}\enspace \mathrm{sin}\enspace {\Phi},\mathrm{cos}\enspace {\Theta}\right),\end{equation*}$$

$$\begin{equation}\boldsymbol{j}\left(w\right)=\frac{1}{2\pi }\enspace {\boldsymbol{e}}_{0}{\times}\left[\enspace \boldsymbol{n}{\times}\boldsymbol{E}\left(w\right)\right].\end{equation} \tag{ 34 }$$

This expression is suitable for calculating the radiated energy in the far-field only, as one should generally have a curl instead of e ₀ and the induced current should not depend on the observer's disposition.

Unlike the surface current density used in the theory of diffraction of plane waves, this one has all three components, including the component perpendicular to the grating surface. This normal component comes about ultimately because the incident electric field has also all three components, unlike the plane wave. For the ultrarelativistic energies, the normal component of the surface current can be safely neglected and in this case the generalize surface current model completely coincides [36] with the well-known approach by Brownell et al [16]. The latter model was successfully tested, for instance, in experiment [37] conducted with a 28.5 GeV electron beam. However for the moderate electron energies, needed for observation of the effects we discuss in this work, the normal component of the surface current is crucially important, which is why we employ the more general model of reference [15].

To calculate the radiation fields at large distances we use equation (28) from [15]

$$\begin{equation}{\boldsymbol{E}}^{R}\approx \frac{i\omega \enspace {\mathrm{e}}^{\mathrm{i}k{r}_{0}}}{2\pi {r}_{0}}\int {\boldsymbol{e}}_{0}{\times}\left[\enspace \boldsymbol{n}{\times}\boldsymbol{E}\left({k}_{x},y,z,\omega \right)\right]{\mathrm{e}}^{-\mathrm{i}{k}_{z}z}\enspace \mathrm{d}z,\end{equation} \tag{ 35 }$$

where the integration is performed along the periodic grating.

In appendix we calculate explicit expressions for the electromagnetic fields produced by a vortex electron [18] in the Cartesian coordinates. One can separate the field into the contributions of the charge, of the magnetic moment, and of the quadrupole moment as follows:

$$\begin{align}\hfill & \boldsymbol{E}\left(\boldsymbol{r},t\right)={\boldsymbol{E}}_{e}\left(\boldsymbol{r},t\right)+{\boldsymbol{E}}_{\mu }\left(\boldsymbol{r},t\right)+{\boldsymbol{E}}_{Q}\left(\boldsymbol{r},t\right),\hfill \\ \hfill & {\boldsymbol{E}}_{e}\left(\boldsymbol{r},t\right)=\frac{1}{{R}^{3}}\left\{\gamma \boldsymbol{\rho },{R}_{z}\right\},\hfill \end{align} \tag{ 36 }$$

$$\begin{align}\hfill & {\boldsymbol{E}}_{\mu }\left(\boldsymbol{r},t\right)=\frac{\ell }{2m}\frac{3\beta \gamma }{{R}^{5}}{R}_{z}\left\{y,-x,0\right\},\hfill \\ \hfill & {\boldsymbol{E}}_{Q}\left(\boldsymbol{r},t\right)=\frac{\gamma }{4{R}^{3}}\boldsymbol{\rho }\left(3\frac{{\bar{\rho }}_{0}^{2}}{{R}^{2}}\left(1-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)+{\ell }^{2}{\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2}\left[3\frac{{T}_{z}^{2}}{{R}^{2}}\left(1-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)+3\frac{{R}_{z}^{2}}{{R}^{2}}-6\beta \frac{{R}_{z}{T}_{z}}{{R}^{2}}-1\right]\right)\hfill \end{align} \tag{ 37 }$$

$$\begin{equation}\frac{\gamma }{4{R}^{3}}\left(\boldsymbol{z}-\boldsymbol{\beta }t\right)\left(3\frac{{\bar{\rho }}_{0}^{2}}{{R}^{2}}\left(3-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)+{\ell }^{2}{\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2}\left[\frac{{T}_{z}^{2}}{{R}^{2}}\left(3-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)+3\frac{{R}_{z}^{2}}{{R}^{2}}-1\right]\right).\end{equation} \tag{ 38 }$$

where following notations are used: $\boldsymbol{\beta }=\left(0,0,\beta \right),\enspace \boldsymbol{z}=\left(0,0,z\right)$,

$$\begin{align}\hfill \boldsymbol{\rho }=\left\{x,y\right\},\quad {R}_{z}{:=}\gamma \left(z-\beta t\right),\quad {T}_{z}{:=}\gamma \left(t-\beta z\right),\\ \hfill \boldsymbol{R}=\left\{\boldsymbol{\rho },\gamma \left(z-\beta t\right)\right\}\enspace ,\quad \gamma ={\left(1-{\beta }^{2}\right)}^{-1/2}.\end{align} \tag{ 39 }$$

We omit the magnetic fields, as to calculate the surface current below we need the electric field only. In the problem of S–P radiation, the grating is supposed to be very long in the transverse direction, so we need the Fourier transform of these fields.

When calculating the Fourier transform of the electric fields produced by the wave-packet

$$\begin{equation*}\boldsymbol{E}\left({q}_{x},y,z,\omega \right)=\int \mathrm{d}x\enspace \mathrm{d}t\enspace \boldsymbol{E}\left(\boldsymbol{r},t\right){\text{e}}^{\text{i}\omega t-\mathrm{i}{q}_{x}x}\end{equation*}$$

the following integral and its derivatives appear

$$\begin{equation}{I}_{\nu }\left({q}_{x},y,z,\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}\mathrm{d}t\underset{-\infty }{\overset{\infty }{\int }}\mathrm{d}x\frac{{\mathrm{e}}^{\mathrm{i}\left(\omega t-{q}_{x}x\right)}}{{\left({x}^{2}+{y}^{2}+{\gamma }^{2}{\left(z-\beta t\right)}^{2}\right)}^{\left(\nu /2\right)}},\quad \nu =3,5,7.\end{equation} \tag{ 40 }$$

We consider three master-integrals to reduce a number of derivatives with respect to parameters, although I₃ alone would be enough to calculate the Fourier transform of all the terms. The master integrals read

$$\begin{equation}{I}_{3}\left({q}_{x},y,z,\omega \right)=\frac{2\pi }{\gamma \beta }\enspace \mathrm{exp}\left(\frac{i\omega z}{\beta }-\mu \vert y\vert \right)\frac{1}{\vert y\vert },\end{equation} \tag{ 41 }$$

$$\begin{equation}{I}_{5}\left({q}_{x},y,z,\omega \right)=\frac{2\pi }{\gamma \beta }\enspace \mathrm{exp}\left(\frac{i\omega z}{\beta }-\mu \vert y\vert \right)\frac{\left(1+\mu \vert y\vert \right)}{3\vert y{\vert }^{3}},\end{equation} \tag{ 42 }$$

$$\begin{equation}{I}_{7}\left({q}_{x},y,z,\omega \right)=\frac{2\pi }{\gamma \beta }\enspace \mathrm{exp}\left(\frac{i\omega z}{\beta }-\mu \vert y\vert \right)\frac{\left(3+3\mu \vert y\vert +{\mu }^{2}\vert y{\vert }^{2}\right)}{15\vert y{\vert }^{5}},\end{equation} \tag{ 43 }$$

where $\mu =\sqrt{\frac{{\omega }^{2}}{{\gamma }^{2}{\beta }^{2}}+{q}_{x}^{2}}$, ν = 3, 5, 7.

All the rest can be obtained by taking derivatives of the corresponding master integral either over t or x

$$\begin{equation}{I}_{\nu ,{t}^{s},{x}^{p}}=\underset{-\infty }{\overset{\infty }{\int }}\mathrm{d}t\underset{-\infty }{\overset{\infty }{\int }}\mathrm{d}x\frac{{t}^{s}{x}^{p}\enspace {\mathrm{e}}^{\mathrm{i}\left(\omega t-{q}_{x}x\right)}}{{\left({x}^{2}+{y}^{2}+{\gamma }^{2}{\left(z-\beta t\right)}^{2}\right)}^{\left(\nu /2\right)}}={i}^{p-s}{\partial }_{\omega }^{s}{\partial }_{{q}_{x}}^{p}{I}_{\nu }\left({q}_{x},y,z,\omega \right).\end{equation} \tag{ 44 }$$

Note that only p = 0 (y and z components of electric field) and p = 1 (x component) cases are required. In particular, electric fields from the charge and the magnetic momentum read

$$\begin{equation}{\boldsymbol{E}}_{e}\left({q}_{x},y,z,\omega \right)=\gamma \left(i{\partial }_{{q}_{x}},y,\left(z+i\beta {\partial }_{\omega }\right)\right)\enspace {I}_{3}\left({q}_{x},y,z,\omega \right),\end{equation} \tag{ 45 }$$

$$\begin{equation}{\boldsymbol{E}}_{\mu }\left({q}_{x},y,z,\omega \right)=\frac{3\ell \beta {\gamma }^{2}}{2m}\left(z+i\beta {\partial }_{\omega }\right)\left(y,-i{\partial }_{{q}_{x}},0\right)\enspace {I}_{5}\left({q}_{x},y,z,\omega \right),\end{equation} \tag{ 46 }$$

which after the differentiation reads

$$\begin{align}\hfill & {\boldsymbol{E}}_{e}\left({q}_{x},y,z,\omega \right)=\frac{2\pi }{\beta }\left(-i{q}_{x},\mathrm{sgn}\left(y\right)\sqrt{{\left(\frac{\omega }{\beta \gamma }\right)}^{2}+{q}_{x}^{2}},-i\frac{\omega }{\beta {\gamma }^{2}}\right)\enspace \frac{\mathrm{exp}\left\{iz\frac{\omega }{\beta }-\vert y\vert \sqrt{{\left(\frac{\omega }{\beta \gamma }\right)}^{2}+{q}_{x}^{2}}\right\}}{\sqrt{{\left(\frac{\omega }{\beta \gamma }\right)}^{2}+{q}_{x}^{2}}},\hfill \\ \hfill & {\boldsymbol{E}}_{\mu }\left({q}_{x},y,z,\omega \right)=-\frac{\ell }{2m}\frac{i2\pi \omega }{\beta \gamma }\left(\mathrm{sgn}\left(y\right)\sqrt{{\left(\frac{\omega }{\beta \gamma }\right)}^{2}+{q}_{x}^{2}},i{q}_{x},0\right)\frac{\mathrm{exp}\left\{iz\frac{\omega }{\beta }-\vert y\vert \sqrt{{\left(\frac{\omega }{\beta \gamma }\right)}^{2}+{q}_{x}^{2}}\right\}}{\sqrt{{\left(\frac{\omega }{\beta \gamma }\right)}^{2}+{q}_{x}^{2}}},\hfill \end{align} \tag{ 47 }$$

where ${q}^{2}={q}_{0}^{2}-{\boldsymbol{q}}^{2}\ne 0$ and for the electron packet whose center is at the distance h from the grating one needs to substitute |y| → |y − h|.

Technically, the Fourier transform of the quadrupole fields follows the same line. Starting from the formula (38), one should substitute x and t variables in the numerator by the differential operators $x\to i{\partial }_{{q}_{x}}$, t → −i∂_ω , acting on the master integrals defined by the denominators, R^−ν/2 → I_ν . Resulting expressions are calculated with the aid of computer algebra and can be found in the public repository [38]. Here we only discuss the general structure of the corresponding expressions. Consider the Fourier transform of a term where R_z = γ(z − βt) enters the numerator

$$\begin{equation*}\int \mathrm{d}x\enspace \mathrm{d}t\enspace f\left(x,y\right)\frac{\underset{z}{\overset{n}{R}}}{{R}^{\frac{\nu }{2}}}{\text{e}}^{\text{i}\omega t-\mathrm{i}{q}_{x}x}={\gamma }^{n}f\left(i{\partial }_{{q}_{x}},y\right){\left(z+i\beta {\partial }_{\omega }\right)}^{n-1}\left(z+i\beta {\partial }_{\omega }\right){\mathrm{e}}^{\mathrm{i}\omega z/\beta }{I}_{f}\left({q}_{x},y,\omega \right).\end{equation*}$$

The commutator [iβ∂_ω , e^iωz/β ] = −z e^iωz/β implies that (z + iβ∂_ω )e^iωz/β   I_f (q_x , y, ω) = e^iωz/β (iβ∂_ω )I_f (q_x , y, ω) and we get the integral where z-variable enters only the exponential factor

$$\begin{equation}\int \mathrm{d}x\enspace \mathrm{d}t\enspace f\left(x,y\right)\frac{\underset{z}{\overset{n}{R}}}{{R}^{\frac{\nu }{2}}}{\text{e}}^{\text{i}\omega t-\mathrm{i}{q}_{x}x}={\gamma }^{n}{\mathrm{e}}^{\mathrm{i}\omega z/\beta }f\left(i{\partial }_{{q}_{x}},y\right){\left(i\beta {\partial }_{\omega }\right)}^{n}{I}_{f}\left({q}_{x},y,\omega \right).\end{equation} \tag{ 48 }$$

Therefore we express t using z and R_z ,

$$\begin{equation*}t-\beta z=-\frac{1}{\beta }\left(z-\beta t\right)+\frac{z}{\beta {\gamma }^{2}}=-\frac{{R}_{z}}{\beta \gamma }+\frac{z}{\beta {\gamma }^{2}}\end{equation*}$$

and rewrite (38) as a sum of (48)-like terms. As a result, the Fourier transform of the quadrupole fields has the following factorized structure:

$$\begin{equation}{\boldsymbol{E}}_{Q}\left({q}_{x},y,z,\omega \right)=\mathrm{exp}\left(iz\frac{\omega }{\beta }\right)\left({\boldsymbol{E}}_{{Q}_{0}}\left({q}_{x},y,\omega \right)+{\boldsymbol{E}}_{{Q}_{1}}\left({q}_{x},y,\omega \right)z+{\boldsymbol{E}}_{{Q}_{2}}\left({q}_{x},y,\omega \right){z}^{2}\right),\end{equation} \tag{ 49 }$$

where a z-dependent plane-wave is multiplied by a second order polynomial in z-variable with coefficients being some functions. Note that the constant term of the polynomial has the leading term proportional to the charge contribution

$$\begin{equation*}\enspace \mathrm{exp}\left(iz\frac{\omega }{\beta }\right){\boldsymbol{E}}_{{Q}_{0}}\left({q}_{x},y,\omega \right)=\frac{{\ell }^{2}{\lambda }_{c}^{2}}{{\bar{\rho }}_{0}^{2}}{\boldsymbol{E}}_{e}\left({q}_{x},y,z,\omega \right)+\cdots \end{equation*}$$

The charge and the magnetic dipole contributions depend on z due to the z-dependent plane-wave factor only. That is the Fourier transform of the total electric field has the same structure as in (49),

$$\begin{equation}\boldsymbol{E}\left({q}_{x},y,z,\omega \right)=\mathrm{exp}\left(iz\frac{\omega }{\beta }\right)\left({\boldsymbol{E}}_{0}\left({q}_{x},y,\omega \right)+{\boldsymbol{E}}_{{Q}_{1}}\left({q}_{x},y,\omega \right)z+{\boldsymbol{E}}_{{Q}_{2}}\left({q}_{x},y,\omega \right){z}^{2}\right),\end{equation} \tag{ 50 }$$

where ${\boldsymbol{E}}_{0}\left({q}_{x},y,\omega \right)={\boldsymbol{E}}_{e}\left({q}_{x},y,\omega \right)+{\boldsymbol{E}}_{\mu }\left({q}_{x},y,\omega \right)+{\boldsymbol{E}}_{{Q}_{0}}\left({q}_{x},y,\omega \right)$. The terms linear and quadratic in z contain the quadrupole contribution only and represent the non-paraxial contributions mentioned earlier. We will use this structure in the next section to perform integration along the grating.

The surface current

$$\begin{equation*}\boldsymbol{j}=\frac{1}{2\pi }\enspace \left(-{E}_{x}{e}_{0y},\enspace {E}_{x}{e}_{0x}+{E}_{z}{e}_{0z},\enspace -{E}_{z}{e}_{0y}\right),\end{equation*}$$

inherits the structure of equation (50)

$$\begin{equation}\boldsymbol{j}\left({k}_{x},y,z,\omega \right)=\mathrm{exp}\left(iz\frac{\omega }{\beta }\right)\left({\boldsymbol{j}}_{0}\left({k}_{x},y,\omega \right)+{\boldsymbol{j}}_{{Q}_{1}}\left({k}_{x},y,\omega \right)\enspace z+{\boldsymbol{j}}_{{Q}_{2}}\left({k}_{x},y,\omega \right)\enspace {z}^{2}\right),\end{equation} \tag{ 51 }$$

where the first term contains all types of the contributions, ${\boldsymbol{j}}_{0}\left({k}_{x},y,\omega \right)={\boldsymbol{j}}_{e}\left({k}_{x},y,\omega \right)+{\boldsymbol{j}}_{\mu }\left({k}_{x},y,\omega \right)+{\boldsymbol{j}}_{{Q}_{0}}\left({k}_{x},y,\omega \right)$, while the next terms are related to the quadrupole contribution only. Note that here k = ω r ₀/r₀ is an on-mass-shell wave vector, k² = ω² − k ² = 0.

Integrating with respect to z-coordinate along the periodic grating we get

$$\begin{equation}\underset{0}{\overset{Nd}{\int }}\mathrm{d}z\left({\boldsymbol{j}}_{0}+{\boldsymbol{j}}_{{Q}_{1}}\enspace z+{\boldsymbol{j}}_{{Q}_{2}}\enspace {z}^{2}\right)\mathrm{exp}\left(iz\left(\frac{w}{\beta }-{k}_{z}\right)\right)\end{equation} \tag{ 52 }$$

$$\begin{equation*}=\left({\boldsymbol{j}}_{0}+{\boldsymbol{j}}_{{Q}_{1}}\left(i{\partial }_{{k}_{z}}\right)+{\boldsymbol{j}}_{{Q}_{2}}{\left(i{\partial }_{{k}_{z}}\right)}^{2}\right)F\left({{\Theta}}_{1}\right),\end{equation*}$$

where

$$\begin{equation}F\left({{\Theta}}_{1}\right)=\sum\limits _{j=0}^{N}\underset{jd}{\overset{jd+a}{\int }}\mathrm{d}z\enspace \mathrm{exp}\left(iz\left(\frac{\omega }{\beta }-{k}_{z}\right)\right)=\frac{2\enspace \mathrm{sin}\left(\frac{a{{\Theta}}_{1}}{2}\right)}{{{\Theta}}_{1}}\frac{\mathrm{sin}\left(\frac{N\enspace \mathrm{d}{{\Theta}}_{1}}{2}\right)}{\mathrm{sin}\left(\frac{\mathrm{d}{{\Theta}}_{1}}{2}\right)}\enspace \mathrm{exp}\left(\frac{i{{\Theta}}_{1}}{2}\left(a+\left(N-1\right)d\right)\right),\end{equation} \tag{ 53 }$$

and we denote

$$\begin{equation*}{{\Theta}}_{1}=\frac{\omega }{\beta }-{k}_{z}.\end{equation*}$$

Here a is a strip width (see figure 1). Note that ${\partial }_{{k}_{z}}=-{\partial }_{{{\Theta}}_{1}}$, and to write the resulting formulas in a compact form we will use the following notations:

$$\begin{equation*}{\partial }_{{{\Theta}}_{1}}F\left({{\Theta}}_{1}\right)={F}^{\prime }\left({{\Theta}}_{1}\right),\hspace{25.0pt}{\partial }_{{{\Theta}}_{1}}^{j}F\left({{\Theta}}_{1}\right)={F}^{\left(j\right)}\left({{\Theta}}_{1}\right).\end{equation*}$$

A standard interference factor due to diffraction on a grating is |F|². As can be seen, the radiation from the charge and the magnetic moment is modulated by the standard interference factor |F|², while the interference of the charge with the quadrupole involves F and its derivatives. As a result, the non-symmetric shape of the electron packet results in a small modification of the S–P dispersion relation, equation (1).

The distribution of the radiated energy over the frequencies and angles,

$$\begin{equation}\frac{{\mathrm{d}}^{2}W}{\mathrm{d}\omega \enspace \mathrm{d}{\Omega}}={r}_{0}^{2}\vert {\boldsymbol{E}}^{R}\vert ,\end{equation} \tag{ 54 }$$

represents a sum of the following terms (cf equation (11)):

$$\begin{equation}\mathrm{d}{W}_{ee}=\frac{{\omega }^{2}}{4{\pi }^{2}}\vert {\boldsymbol{j}}_{e}{\vert }^{2}\vert F\left({{\Theta}}_{1}\right){\vert }^{2},\end{equation} \tag{ 55 }$$

$$\begin{equation}\mathrm{d}{W}_{e\mu }=\frac{{\omega }^{2}}{4{\pi }^{2}}\left[{\boldsymbol{j}}_{e}\enspace {\boldsymbol{j}}_{\mu }^{{\ast}}+{\boldsymbol{j}}_{e}^{{\ast}}{\boldsymbol{j}}_{\mu }\right]\vert F\left({{\Theta}}_{1}\right){\vert }^{2},\end{equation} \tag{ 56 }$$

$$\begin{equation}\mathrm{d}{W}_{e{Q}_{0}}=\frac{{\omega }^{2}}{4{\pi }^{2}}\left[{\boldsymbol{j}}_{e}\enspace {\boldsymbol{j}}_{{Q}_{0}}^{{\ast}}+{\boldsymbol{j}}_{e}^{{\ast}}{\boldsymbol{j}}_{{Q}_{0}}\right]\vert F\left({{\Theta}}_{1}\right){\vert }^{2},\end{equation} \tag{ 57 }$$

$$\begin{equation}\mathrm{d}{W}_{e{Q}_{1}}=\frac{i{\omega }^{2}}{4{\pi }^{2}}\left[i{\boldsymbol{j}}_{e}\enspace {\boldsymbol{j}}_{{Q}_{1}}^{{\ast}}F\left({{\Theta}}_{1}\right){F}^{\prime }{\left({{\Theta}}_{1}\right)}^{{\ast}}-{\boldsymbol{j}}_{e}^{{\ast}}{\boldsymbol{j}}_{{Q}_{1}}F{\left({{\Theta}}_{1}\right)}^{{\ast}}{F}^{\prime }\left({{\Theta}}_{1}\right)\right],\end{equation} \tag{ 58 }$$

$$\begin{equation}\mathrm{d}{W}_{e{Q}_{2}}=-\frac{{\omega }^{2}}{4{\pi }^{2}}\left[{\boldsymbol{j}}_{e}\enspace {\boldsymbol{j}}_{{Q}_{2}}^{{\ast}}F\left({{\Theta}}_{1}\right){F}^{{\prime\prime}}{\left({{\Theta}}_{1}\right)}^{{\ast}}+{\boldsymbol{j}}_{e}^{{\ast}}{\boldsymbol{j}}_{{Q}_{2}}F{\left({{\Theta}}_{1}\right)}^{{\ast}}{F}^{{\prime\prime}}\left({{\Theta}}_{1}\right)\right],\end{equation} \tag{ 59 }$$

$$\begin{equation}\mathrm{d}{W}_{\mu {Q}_{j}}=-\frac{{\left(-i\right)}^{j}{\omega }^{2}}{4{\pi }^{2}}\left[{\boldsymbol{j}}_{\mu }{\boldsymbol{j}}_{{Q}_{j}}^{{\ast}}F\left({{\Theta}}_{1}\right){F}^{\left(j\right)}{\left({{\Theta}}_{1}\right)}^{{\ast}}-{\boldsymbol{j}}_{\mu }^{{\ast}}{\boldsymbol{j}}_{{Q}_{j}}F{\left({{\Theta}}_{1}\right)}^{{\ast}}{F}^{\left(j\right)}\left({{\Theta}}_{1}\right)\right],\end{equation} \tag{ 60 }$$

$$\begin{equation}\mathrm{d}{W}_{\mu \mu }=\frac{{\omega }^{2}}{4{\pi }^{2}}\vert {\boldsymbol{j}}_{\mu }{\vert }^{2}\vert F\left({{\Theta}}_{1}\right){\vert }^{2},\end{equation} \tag{ 61 }$$

$$\begin{equation}\mathrm{d}{W}_{{Q}_{j}{Q}_{k}}=\frac{{\left(-i\right)}^{j+k}{\omega }^{2}}{4{\pi }^{2}}\left[{\left(-1\right)}^{k}{\boldsymbol{j}}_{{Q}_{j}}{\boldsymbol{j}}_{{Q}_{k}}^{{\ast}}{F}^{\left(j\right)}\left({{\Theta}}_{1}\right){F}^{\left(k\right)}{\left({{\Theta}}_{1}\right)}^{{\ast}}+{\left(-1\right)}^{j}{\boldsymbol{j}}_{{Q}_{j}}^{{\ast}}{\boldsymbol{j}}_{{Q}_{k}}{F}^{\left(j\right)}{\left({{\Theta}}_{1}\right)}^{{\ast}}{F}^{\left(k\right)}\left({{\Theta}}_{1}\right)\right].\end{equation} \tag{ 62 }$$

From here, the charge radiation and the charge-dipole interference can be explicitly calculated

$$\begin{align}\hfill & \frac{{\mathrm{d}}^{2}{W}_{ee}}{\mathrm{d}\omega \enspace \mathrm{d}{\Omega}}=\mathrm{exp}\left(-\frac{2y}{{h}_{\text{eff}}}\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}\right)\hfill \\ \hfill & \qquad {\times}\frac{{\mathrm{cos}}^{2}\enspace {\Theta}+2\beta {\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace \mathrm{cos}\enspace {\Theta}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}+{\mathrm{sin}}^{2}\enspace {\Phi}{\mathrm{sin}}^{2}\enspace {\Theta}+{\beta }^{2}{\gamma }^{4}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{4}\enspace {\Theta}}{{\gamma }^{2}{\left(1-\beta \enspace \mathrm{cos}\enspace {\Theta}\right)}^{2}\left(1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}\right)}\enspace \vert F{\vert }^{2},\hfill \end{align} \tag{ 63 }$$

$$\begin{align}\hfill & \frac{{\mathrm{d}}^{2}{W}_{e\mu }}{\mathrm{d}\omega \enspace \mathrm{d}{\Omega}}=\frac{\ell }{m}\enspace \mathrm{exp}\left(-\frac{2y}{{h}_{\text{eff}}}\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}\right)\frac{\omega \enspace \mathrm{cos}\enspace {\Phi}\enspace \mathrm{sin}\enspace {\Theta}\left(\beta {\gamma }^{2}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}+\mathrm{cos}\enspace {\Theta}\right)}{{\gamma }^{2}{\left(1-\beta \enspace \mathrm{cos}\enspace {\Theta}\right)}^{2}\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}}\enspace \vert F{\vert }^{2}.\hfill \end{align} \tag{ 64 }$$

The charge contribution reproduces ⁵ equation (56) of [36]. The charge-dipole interference results in an azimuthal asymmetry arising from cos Φ, analogously to other types of polarization radiation [39]. As noted earlier, the eμ contribution vanishes at Φ = π/2.

The quadrupole contributions from ${\boldsymbol{j}}_{{Q}_{1}}$ and ${\boldsymbol{j}}_{{Q}_{2}}$ are defined by real parts of the product of currents, by the form factor F and its derivatives. Nevertheless, explicit calculations show ⁶ that these terms have also a factorized structure

$$\begin{equation}\mathrm{d}{W}_{e{Q}_{j}}=\mathrm{exp}\left(-\frac{2y}{{h}_{\text{eff}}}\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}\right){P}_{e{Q}_{j}}\left({k}_{x},y,\omega \right){F}_{e{Q}_{j}}\left(\omega ,{k}_{z}\right).\end{equation} \tag{ 65 }$$

Here, the functions ${P}_{e{Q}_{j}}\left({k}_{x},y,\omega \right)$ define the angular distributions and ${F}_{e{Q}_{j}}\left(\omega ,{k}_{z}\right)$ determine positions of the spectral lines and their width (therefore we will call ${F}_{e{Q}_{j}}\left(\omega ,{k}_{z}\right)$ a spectral factor). In figure 2 we compare the radiation intensities normalized per 1 strip from gratings with N = 25 and N = 50 strips. The spectral curves of dW_ee , $\mathrm{d}{W}_{e{Q}_{0}}$ and $\mathrm{d}{W}_{e{Q}_{2}}$ have a similar shape and position; up to the sign this is also the case for dW_eμ , which is zero at Φ = π/2 by the symmetry considerations. The contribution $\mathrm{d}{W}_{e{Q}_{1}}$ leads to a shift of the spectral line, but its amplitude is rather small (a factor of 10² is used in figure 2) and this shift is almost unobservable. A nonlinear amplification of the quadrupole contribution to the radiation intensity is clearly seen in figure 2. This effect becomes stronger for the radiation in the forward direction (see figure 3).

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Zoom In Zoom Out Reset image size

Figures 2 and 3 correspond to the case when both number of strips N = 50 and OAM, ℓ = 1000 are close to maximal values estimated in section 2.2. Table 1 gives the corresponding dimensionless parameters. Note that two higher order terms ${\eta }_{{Q}_{12}}$ and ${\eta }_{{Q}_{22}}$ also surpass quantum recoil in this case. Moreover, ${\eta }_{{Q}_{22}}$, being two orders of magnitude smaller than the leading correction ${\eta }_{{Q}_{2}}$, becomes more important than ${\eta }_{{Q}_{0}}$ and ${\eta }_{{Q}_{1}}$ corrections. This means that within our perturbative method, only charge and charge-quadrupole ${\eta }_{{Q}_{2}}$ contributions should be computed, while all the rest corrections can be considered as NLO order corrections which are at least two orders of magnitude smaller (we plot the corresponding curves in figures 2 and 3 just to demonstrate their shapes).

Table 1. Dimensionless parameters of the model which correspond to the figure 2. The dynamical non-paraxial contribution η_Q2 is the biggest one.

ηq = ω/ɛ	ημ = ℓλc /λ	${\eta }_{Q\mathrm{0}}={\bar{\rho }}_{0}^{2}/{h}_{\text{eff}}^{2}$	${\eta }_{Q\mathrm{1}}={\ell }^{2}{\lambda }_{c}^{2}/{\bar{\rho }}_{0}^{2}$	${\eta }_{Q\mathrm{2}}={N}^{2}\enspace {\ell }^{2}{\lambda }_{c}^{2}/{\bar{\rho }}_{0}^{2}$	N
1.95 × 10−10	1.95 × 10−7	2.67 × 10−6	1.69 × 10−6	4.22 × 10−3	50
${\eta }_{\mu \mu }={\eta }_{\mu }^{2}$	${\eta }_{\mu {Q}_{0}}={\eta }_{\mu }{\eta }_{{Q}_{0}}$	${\eta }_{\mu {Q}_{1}}={\eta }_{\mu }{\eta }_{{Q}_{1}}$	${\eta }_{\mu {Q}_{2}}={\eta }_{\mu }{\eta }_{{Q}_{2}}$	ℓ	${\bar{\rho }}_{0}\enspace ,\mu \text{m}$
3.8 × 10−14	5.2 × 10−13	3.29 × 10−13	8.23 × 10−10	1000	0.3
${\eta }_{{Q}_{00}}={\eta }_{{Q}_{0}}{\eta }_{{Q}_{0}}$	${\eta }_{{Q}_{01}}={\eta }_{{Q}_{0}}{\eta }_{{Q}_{1}}$	${\eta }_{{Q}_{02}}={\eta }_{{Q}_{0}}{\eta }_{{Q}_{2}}$	${\eta }_{{Q}_{11}}={\eta }_{{Q}_{1}}{\eta }_{{Q}_{1}}$	${\eta }_{{Q}_{12}}={\eta }_{{Q}_{1}}{\eta }_{{Q}_{2}}$	${\eta }_{{Q}_{22}}={\eta }_{{Q}_{2}}{\eta }_{{Q}_{2}}$
7.11 × 10−12	4.5 × 10−12	1.13 × 10−8	2.85 × 10−12	7.13 × 10−9	1.78 × 10−5

Studies of the radiation from classical beams show that horizontal and vertical beam spreading lead to some modifications of the spectral line [40]. The horizontal spreading of the beam shifts the spectral line towards lower frequencies while the vertical spreading results in the opposite shift. A combination of both spreading types results in a broadening of the spectral line. Here we show that quantum coherence of the wave packet may lead to a different behavior. Namely, despite the vertical–horizontal spreading of the wave-packet, the resulting spectral line does not demonstrate a broadening until the quadrupole–quadrupole corrections come into play, which is the case for long gratings with N ≫ N_max only.

Such a stabilization of the line width can be explained using (58), (59) and properties of the function F(ω). First of all, instead of a full width at half maximum (FWHM) one can consider a full width between zeros of the spectral curves. Zeros of the interference contributions are defined by zeros of the kernel F itself ⁷ . Therefore, all contributions (55)–(61) have the same full width between zeros of the spectral curves (in figure 2 an example of these coinciding zeros near the spectral maximum is presented). This strongly restricts the possible broadening of the spectral line and at large N all contributions (55)–(61) tend to have the same width.

The quadrupole–quadrupole corrections $\mathrm{d}{W}_{{Q}_{1}{Q}_{2}}$ and $\mathrm{d}{W}_{{Q}_{2}{Q}_{2}}$ contain only derivatives of the kernel F. As a result, zeros of their spectral curves (in a vicinity of the maximum) disappear and the corresponding lines demonstrate a broadening (see figure 4). Importantly, if one takes into account these contributions then the next corrections of the same order should also be taken into account, such as interference of the charge with the octupole magnetic moment, with 16 pole electric moment and so forth (see (10)). However, the octupole magnetic moment contribution has the same symmetry as the magnetic momentum contribution, thus vanishing in the zenith direction. Regarding dW_e16p contribution, because it is the interference term the zeros of the kernel F will also prevent a broadening of the corresponding spectral line. As a result, the next corrections that may lead to a broadening are only the quadrupole–quadrupole ones. Therefore figure 4 and table 2 contain all necessary terms.

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Numerical studies of the spectral lines reveal not only an absence of the broadening, but even a slight narrowing of the lines due to the charge-quadrupole interference. The FWHM for various grating lengths are presented in table 2 where a narrowing of the line (charge + LO corrections column) can be seen. When N > 100 the quadrupole–quadrupole contributions become important (charge + LO + NLO corrections column of table 2) and when N > 150 broadening due to the horizontal–vertical spreading surpasses the narrowing. Thus, for N < 100 we can safely compute $\mathrm{d}W=\mathrm{d}{W}_{e}+\mathrm{d}{W}_{e{Q}_{2}}$ for parameters from table 1.

A physical reason for the line narrowing is also spreading of the wave packet. Indeed, the natural width of the spectral line Δω is related to the time scale of the radiation process Δt by a following uncertainty relation:

$$\begin{equation}{\Delta}\omega \sim \frac{1}{{\Delta}t}.\end{equation} \tag{ 66 }$$

Due to the packet spreading, its interaction with the strips lasts longer, especially at the end of the long grating with dN_max ≫ z_R , and so Δt(z) grows. In other words, the spreading slightly increases the radiation formation length.

Let us denote

$$\begin{equation*}{\omega }_{g}=\frac{2\pi }{{\lambda }_{g}}=\frac{2\pi \enspace g}{d\left({\beta }^{-1}-\mathrm{cos}\enspace {\Theta}\right)},\quad g=1,2,3,\dots \end{equation*}$$

then |F|² contains a Fejér kernel

$$\begin{align}\hfill {F}_{N}\left(\omega \right)& =\frac{{\mathrm{sin}}^{2}\left(\frac{N\enspace \mathrm{d}{{\Theta}}_{1}}{2}\right)}{N\enspace {\mathrm{sin}}^{2}\left(\frac{\mathrm{d}{{\Theta}}_{1}}{2}\right)}\underset{N\to \infty }{\to }2\pi \sum\limits _{g}\delta \left(\mathrm{d}\omega \left(1/\beta -\mathrm{cos}\enspace {\Theta}\right)-\mathrm{d}{\omega }_{g}\left(1/\beta -\mathrm{cos}\enspace {\Theta}\right)\right)\hfill \\ \hfill & \hfill =\sum\limits _{g}\frac{2\pi g}{g\mathrm{d}\left({\beta }^{-1}-\mathrm{cos}\enspace {\Theta}\right)}\delta \left(\omega -{\omega }_{g}\right)=\sum\limits _{g}\frac{{\omega }_{g}}{g}\delta \left(\omega -{\omega }_{g}\right)\end{align} \tag{ 67 }$$

which can be used to integrate over frequencies in a vicinity of the resonant one. For the charge, the charge-dipole and the charge-Q₀ contributions, the Fejér kernel can be substituted by a delta function when N is large. For a grating of finite length with N strips the spectral line has a width proportional to 1/N. The angular distributions of the charge radiation and of the charge-dipole radiation for the main diffraction order g = 1 read

$$\begin{align}\hfill & \frac{\mathrm{d}{W}_{ee}}{\mathrm{d}{\Omega}}=N\frac{{d}^{2}{\omega }_{1}^{3}}{{\pi }^{2}}\enspace {\mathrm{sin}}^{2}\left(\frac{a\pi }{d}\right)\mathrm{exp}\left(-\frac{2{\omega }_{1}y}{\beta \gamma }\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}\right)\hfill \\ \hfill & \qquad {\times}\frac{{\mathrm{cos}}^{2}{\Theta}+2\beta {\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace \mathrm{cos}\enspace {\Theta}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}+{\mathrm{sin}}^{2}\enspace {\Phi}{\mathrm{sin}}^{2}\enspace {\Theta}+{\beta }^{2}{\gamma }^{4}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{4}\enspace {\Theta}}{{\beta }^{2}{\gamma }^{2}\left(1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}\right)},\hfill \end{align} \tag{ 68 }$$

$$\begin{align}\hfill & \frac{\mathrm{d}{W}_{e\mu }}{\mathrm{d}{\Omega}}=N\frac{\ell {d}^{2}{\omega }_{1}^{4}}{m{\pi }^{2}}\enspace \frac{1}{{\beta }^{2}{\gamma }^{2}}\enspace {\mathrm{sin}}^{2}\left(\frac{\pi a}{d}\right)\mathrm{exp}\left(-\frac{2{\omega }_{1}y}{\beta \gamma }\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}\right)\hfill \\ \hfill & \qquad {\times}\frac{\mathrm{cos}\enspace {\Phi}\enspace \mathrm{sin}\enspace {\Theta}\left(\beta {\gamma }^{2}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}+\mathrm{cos}\enspace {\Theta}\right)}{\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}}.\hfill \end{align} \tag{ 69 }$$

Both the intensities linearly increase with N.

Integration of the non-paraxial terms, $\mathrm{d}{W}_{e{Q}_{1}}$ and $\mathrm{d}{W}_{e{Q}_{2}}$, is more tricky. First we note that at large N the spectral factors ${F}_{e{Q}_{j}}$ are concentrated near the S–P frequency and have a width ∼1/N. ${F}_{e{Q}_{1}}$ is approximately an odd function and ${F}_{e{Q}_{2}}$ is an even function of ω − ω₁ (see figure 2). Therefore ${F}_{e{Q}_{1}}$ produces a shift of the spectral maximum, whereas ${F}_{e{Q}_{2}}$ amplifies the intensity. At large N these spectral factors are related with the Fejér kernel and its derivatives.

For instance, the charge-Q₂ intensity has the following factor at large N:

$$\begin{equation}{F}_{e{Q}_{2}}=2{d}^{2}{N}^{3}{{\Theta}}_{1}^{2}\enspace {\mathrm{sin}}^{2}\left[\frac{a{{\Theta}}_{1}}{2}\right]{F}_{N}\left(\omega \right)+\mathrm{O}\left(N\right)\end{equation} \tag{ 70 }$$

which is just proportional to the Fejér kernel and can be substituted by a delta-function. As a result, the dynamically enhanced charge-quadrupole interference term $\mathrm{d}{W}_{e{Q}_{2}}/\mathrm{d}{\Omega}$ reads

$$\begin{equation}\frac{\mathrm{d}{W}_{e{Q}_{2}}}{\mathrm{d}{\Omega}}={\ell }^{2}{\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2}\enspace \frac{1}{3{\beta }^{4}{\gamma }^{4}}\enspace \frac{{f}_{2}\left(N\right)}{{\lambda }_{1}^{2}}\enspace \frac{\mathrm{d}{W}_{ee}}{\mathrm{d}{\Omega}}.\end{equation} \tag{ 71 }$$

Expectedly, this dynamical contribution is suppressed in the relativistic case, γ ≫ 1, when the spreading is marginal. Here

$$\begin{align}\hfill & {f}_{2}\left(N\right)=3\pi ad\enspace \mathrm{cot}\left(\frac{a\pi }{d}\right)+3{\pi }^{2}\enspace {a}^{2}+3\pi \enspace ad\left(N-1\right)+{d}^{2}\enspace \left({\pi }^{2}\left(2{N}^{2}-3N+1\right)-3\right)\approx \hfill \\ \hfill & \approx 2{\pi }^{2}{d}^{2}{N}^{2}\quad \text{when}\enspace N\gg 1.\hfill \end{align} \tag{ 72 }$$

Note that

$$\begin{align}\hfill & \frac{{f}_{2}\left(N\right)}{{\lambda }_{1}^{2}}\approx \frac{2{\pi }^{2}{d}^{2}{N}^{2}}{{\lambda }_{1}^{2}}\sim 2{\pi }^{2}{N}^{2}\quad \text{at}\enspace N\gg 1,\hfill \\ \hfill & \quad \text{as}\enspace {\lambda }_{1}\sim d\enspace \text{everywhere}\;\text{except}\enspace {\Theta}\to 0,\hfill \end{align} \tag{ 73 }$$

which leads to a cubic growth for the quadrupole-charge contribution $\mathrm{d}{W}_{e{Q}_{2}}/\mathrm{d}{\Omega}$ with respect to the number of strips ($\mathrm{d}{W}_{e{Q}_{2}}/\mathrm{d}{\Omega}\sim {N}^{2}\enspace \mathrm{d}{W}_{ee}/\mathrm{d}{\Omega}$, dW_ee /dΩ ∼ N). The ratio of this correction to the radiation of the charge also has a non-linear (quadratic) N-dependence

$$\begin{equation*}\frac{\mathrm{d}{W}_{e{Q}_{2}}}{\mathrm{d}{\Omega}}/\frac{\mathrm{d}{W}_{ee}}{\mathrm{d}{\Omega}}\sim {\ell }^{2}\enspace {N}^{2}\enspace {\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2},\quad N\gg 1.\end{equation*}$$

Importantly, $1/{\lambda }_{1}^{2}\left({\Theta}\right)$ is the only additional angle-dependent factor in $\mathrm{d}{W}_{e{Q}_{2}}$ compared to dW_ee , so the dynamical contribution is increased for smaller wavelengths—that is, for smaller emission angles, Θ → 0. Namely, at β ≈ 0.5 we have

$$\begin{equation}\frac{\mathrm{d}{W}_{e{Q}_{2}}\left({\Theta}=0\right)}{\mathrm{d}{W}_{e{Q}_{2}}\left({\Theta}=\pi /2\right)}\approx 4.\end{equation} \tag{ 74 }$$

A large N asymptotic of the charge-Q₁ spectral factor reads

$$\begin{equation}{F}_{e{Q}_{1}}=-N\omega \enspace {\mathrm{sin}}^{2}\left[\frac{a{{\Theta}}_{1}}{2}\right]{F}_{N}^{\prime }\left(\omega \right)+O\left(1\right),\end{equation} \tag{ 75 }$$

where a derivative of the Fejér kernel appears

$$\begin{equation*}{F}_{N}^{\prime }\left(\omega \right)=\frac{\mathrm{d}{{\Theta}}_{1}\enspace \mathrm{sin}\left(\frac{\mathrm{d}N{{\Theta}}_{1}}{2}\right)\left[\mathrm{cot}\left(\frac{\mathrm{d}{{\Theta}}_{1}}{2}\right)\mathrm{sin}\left(\frac{\mathrm{d}N{{\Theta}}_{1}}{2}\right)-N\enspace \mathrm{cos}\left(\frac{\mathrm{d}N{{\Theta}}_{1}}{2}\right)\right]}{N\omega \enspace {\mathrm{sin}}^{2}\left(\frac{\mathrm{d}{{\Theta}}_{1}}{2}\right)}\underset{N\to \infty }{\to }\sum \frac{{\omega }_{g}}{g}{\delta }^{\prime }\left(\omega -{\omega }_{g}\right).\end{equation*}$$

Integration of (65) when g = 1 can be done using a substitution of F'_N (ω) by a derivative of a delta-function

$$\begin{equation*}\frac{\mathrm{d}{W}_{e{Q}_{1}}}{\mathrm{d}{\Omega}}=-N{\omega }_{1}\int \mathrm{exp}\left(\frac{-2y\omega }{\beta \gamma }\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}\right){P}_{e{Q}_{j}}\left({k}_{x},y,\omega \right)\omega \enspace {\mathrm{sin}}^{2}\left[\frac{a{{\Theta}}_{1}}{2}\right]{\delta }^{\prime }\left(\omega -{\omega }_{g}\right)\mathrm{d}\omega \end{equation*}$$

$$\begin{equation*}=N{\omega }_{1}{\partial }_{\omega }{\left.\left[\mathrm{exp}\left(\frac{-2y\omega }{\beta \gamma }\sqrt{1+{\beta }^{2}{\gamma }^{2}\enspace {\mathrm{cos}}^{2}\enspace {\Phi}\enspace {\mathrm{sin}}^{2}\enspace {\Theta}}\right){P}_{e{Q}_{j}}\left({k}_{x},y,\omega \right)\omega \enspace {\mathrm{sin}}^{2}\left(\frac{a{{\Theta}}_{1}}{2}\right)\right]\right\vert }_{\omega \to {\omega }_{1}}.\end{equation*}$$

Using explicit expressions of the radiation intensities one can isolate the dimensionless parameters related with the quadrupole contribution. From equation (71), one can find ${\eta }_{{Q}_{2}}$, while ${\eta }_{{Q}_{1}}$ and ${\eta }_{{Q}_{2}}$ can be extracted from the ratios $\mathrm{d}{W}_{e{Q}_{j}}/\mathrm{d}{W}_{ee}$:

$$\begin{equation}\frac{\mathrm{d}{W}_{e{Q}_{0}}}{\mathrm{d}{W}_{ee}}=\frac{{\bar{\rho }}_{0}^{2}}{{h}_{\text{eff}}^{2}}{P}_{1}\left(\beta ,{\Theta},{\Phi}\right)+{\ell }^{2}\frac{{\lambda }_{c}^{2}}{{\bar{\rho }}_{0}^{2}}{P}_{2}\left(\beta ,\frac{h}{d},{\Theta},{\Phi}\right),\end{equation} \tag{ 76 }$$

$$\begin{equation}\frac{\mathrm{d}{W}_{e{Q}_{1}}}{\mathrm{d}{W}_{ee}}={\ell }^{2}\frac{{\lambda }_{c}^{2}}{{\bar{\rho }}_{0}^{2}}{P}_{3}\left(\beta ,\frac{h}{d},{\Theta},{\Phi}\right),\end{equation} \tag{ 77 }$$

where P_1,2,3 are some smooth functions. One can identify three parameters (16) and (17). Note that $\frac{\mathrm{d}{W}_{e{Q}_{0}}}{\mathrm{d}{W}_{ee}}$ contains a linear combination of independent parameters ${\eta }_{{Q}_{0}}$ and ${\eta }_{{Q}_{1}}$, which we split for a convenience. For ultra-relativistic energies, all corrections from the quadrupole radiation are suppressed.

In figure 5 we plot an azimuthal distribution of the radiation intensity at λ = 2 mm and compare different contributions to the total radiation intensity. We fix the impact parameter and initial radius of the wave packet and consider two cases: ℓ = 1000, N = 50 and ℓ = 100, N = 500. In both cases the maximal number of strips for a given impact parameter, OAM and the initial radius ${\bar{\rho }}_{0}$ is used. A larger grating length corresponds to the smaller angular momentum ℓ = 100. A scaling invariance ℓ → ℓ, N → ⁻¹ N of the quadrupole correction $\mathrm{d}{W}_{e{Q}_{2}}/\mathrm{d}{W}_{ee}\sim {\eta }_{{Q}_{2}}$ can be observed in figure 5. In other words, large OAM lead to a quick spreading and require short gratings, while small OAM result in a relatively slow spreading and allow one to use longer gratings. At the same time, other corrections—in particular from the magnetic moment—depend on ℓ only. Their observation requires the largest possible OAM (ℓ ∼ 10³ and higher) and Φ ≠ π/2.

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Note that for β = 0.5, the charge and the charge-quadrupole contributions have almost the same azimuthal dependence (which is defined mostly by the exponential factor). The charge-magnetic moment contribution yields the small azimuthal asymmetry. The analysis of reference [2] seems reasonable for the case of the S–P radiation too. In the case of THz radiation (λ ∼ 1 mm) this effect is almost unobservable (see figure 5 and table 1). An asymmetry of the order of 0.1% can be seen for infrared S–P radiation, λ ∼ 1 μm, which could in principle be measured.

Equations (71) and (72) show that $\mathrm{d}{W}_{e{Q}_{2}}$ contribution has a cubic growth with the number of strips compared to the linear growth of dW_ee . This is due to constructive interference of the quadrupole radiation from each strip, taken into account that the quadrupole moment is increased (quadratically) because of the spreading. Recall that the maximal grating length (interaction length) and the number of strips are limited by (5) to guarantee that the mean wave packet radius $\bar{\rho }\left(t\right)$ stays smaller than the impact parameter h.

A large impact parameter h should be chosen to obtain a large N_max, therefore we consider ${\bar{\rho }}_{0}/h\ll 1$ and thus approximately ${N}_{\mathrm{max}}^{3}\propto {h}^{3}$. At the same time, the radiation intensity decreases exponentially with the large impact parameters. Taking the maximal number of strips, the dependence of the charge-quadrupole interference term $\mathrm{d}{W}_{e{Q}_{2}}$ on the impact parameter reads

$$\begin{equation}\mathrm{d}{W}_{e{Q}_{2}}\left[\rho ,\ell ,h,{N}_{\mathrm{max}}\left(\rho ,\ell ,h\right)\right]\propto \frac{{\bar{\rho }}_{0}}{\ell }\left(\frac{d}{{\lambda }_{c}{h}_{\text{eff}}^{5}}\right){h}^{3}\enspace {\mathrm{e}}^{-\frac{2h}{{h}_{\text{eff}}}}.\end{equation} \tag{ 78 }$$

The maximum of this contribution defines the optimal impact parameter

$$\begin{equation}{h}_{\text{opt}}\approx \frac{3}{2}\enspace {h}_{\text{eff}}\sim {h}_{\text{eff}}.\end{equation} \tag{ 79 }$$

Note that $\mathrm{d}{W}_{e{Q}_{2}}$ is proportional to ${\bar{\rho }}_{0}/\ell$ when N takes its maximal value (5). That is, a wide packet with a small OAM (recall the corresponding lower bound (28)) can be chosen to simplify experimental studies of the S–P radiation from the vortex electrons.

In figure 6 the behavior of the radiation intensity is shown for the optimal value of h and N_max (h) = 700. Two cases of OAM, ℓ = 150 and ℓ = 100, were considered. The maximal number of strips was calculated for the larger OAM. The diffraction time is inversely proportional to ℓ, therefore a grating optimal for a wave packet with ℓ = 150 can be used also in the case of ℓ < 150. In this case a factor ℓ²/150² reduces the charge-quadrupole radiation intensity $\mathrm{d}{W}_{e{Q}_{2}}$.

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The non-linear dependence of the S–P radiation on the number of strips or on the length of the grating due to the increasing quadrupole moment is clearly seen in figure 6. If observed experimentally, such a non-linear dependence would serve as a hallmark for a non-paraxial regime of electromagnetic radiation, in which an electron packet emits photons as if its charge were smeared over all the coherence length, somewhat similar to a multi-particle beam but with a total charge e. Another possibility to detect this effect is to change OAM for the same diffraction grating and the same scattering geometry and to study corresponding modifications of the radiation intensity.

An additional possibility to detect the charge-quadrupole contribution follows from a polar dependence in (71). In most cases, the total radiation intensity is approximately $\mathrm{d}W=\mathrm{d}{W}_{e}+\mathrm{d}{W}_{e{Q}_{2}}$ because other corrections are at least 2 orders of magnitude smaller than $\mathrm{d}{W}_{e{Q}_{2}}$. In the case of large N and ${\Phi}=\frac{\pi }{2}$, the maximum of the radiation intensity with respect to the polar angle Θ can be found by maximization of the following expression:

$$\begin{equation}\mathrm{d}W\propto {\mathrm{e}}^{\frac{-2h}{{h}_{\text{eff}}}}\left({h}_{\text{eff}}^{-3}+\delta {h}_{\text{eff}}^{-5}\right)\enspace ,\quad \delta =\frac{{\eta }_{{Q}_{2}}}{6}\frac{{d}^{2}}{{\gamma }^{2}{\beta }^{2}}.\end{equation} \tag{ 80 }$$

To begin with the polar angle of maximal intensity for the charge radiation Θ_e we just put δ = 0 which gives h_eff(Θ_e ) = 2h/3 from a linear equation. A non-zero δ leads to the cubic equation. We use Cardano's formula and assume δ small to calculate the first order correction

$$\begin{equation}{h}_{\text{eff}}\left({\Theta}\right)={h}_{\text{eff}}\left({{\Theta}}_{e}\right)-\frac{\delta }{h}.\end{equation} \tag{ 81 }$$

Next, we write effective impact parameters explicitly in terms of the polar angle

$$\begin{equation*}\frac{\beta \gamma d}{2\pi }\left(\mathrm{cos}\left({{\Theta}}_{e}\right)-\mathrm{cos}\left({{\Theta}}_{e}+\delta {\Theta}\right)\right)=-\frac{\delta }{h}\end{equation*}$$

and for small shifts of the maximum, cos δΘ ∼ 1, sin δΘ ∼ δΘ we get an estimate

$$\begin{equation}\delta {\Theta}=-{\eta }_{{Q}_{2}}\frac{\pi d}{3h{\gamma }^{3}{\beta }^{3}\enspace \mathrm{sin}\left({{\Theta}}_{e}\right)}.\end{equation} \tag{ 82 }$$

In figure 7 we plot an example of this effect. The approximate shifts δΘ(N = 800) = −4.64°, δΘ(N = 400) = −1.16° from (82) and numerical calculations δΘ(N = 800) = −4.48°, δΘ(N = 400) = −1.15° are in a good agreement.

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Consider the vortex electron described by the LG packet (2) with n = 0. Its electromagnetic fields represent a sum of those of the charge e, of the magnetic moment μ , and of the electric quadrupole moment Q_αβ , (9). The fields in cylindrical coordinates in the rest frame were calculated in [18]. In our problem, we prefer to use the Cartesian coordinates:

$$\begin{align}\hfill & {E}_{x}=\frac{x}{{r}^{3}}\left[1+\frac{1}{4}\left(\frac{3{\bar{\rho }}_{0}^{2}}{{r}^{2}}\left(1-5\frac{{z}^{2}}{{r}^{2}}\right)+\frac{{l}^{2}{\lambda }_{C}^{2}}{{\bar{\rho }}_{0}^{2}}\left(\frac{3{t}^{2}}{{r}^{2}}\left(1-5\frac{{z}^{2}}{{r}^{2}}\right)+3\frac{{z}^{2}}{{r}^{2}}-1\right)\right)\right],\hfill \\ \hfill & {E}_{y}=\frac{y}{{r}^{3}}\left[1+\frac{1}{4}\left(\frac{3{\bar{\rho }}_{0}^{2}}{{r}^{2}}\left(1-5\frac{{z}^{2}}{{r}^{2}}\right)+\frac{{l}^{2}{\lambda }_{C}^{2}}{{\bar{\rho }}_{0}^{2}}\left(\frac{3{t}^{2}}{{r}^{2}}\left(1-5\frac{{z}^{2}}{{r}^{2}}\right)+3\frac{{z}^{2}}{{r}^{2}}-1\right)\right)\right],\hfill \\ \hfill & {E}_{z}=\frac{z}{{r}^{3}}\left[1+\frac{1}{4}\left(\frac{3{\bar{\rho }}_{0}^{2}}{{r}^{2}}\left(3-5\frac{{z}^{2}}{{r}^{2}}\right)+\frac{{l}^{2}{\lambda }_{C}^{2}}{{\bar{\rho }}_{0}^{2}}\left(\frac{3{t}^{2}}{{r}^{2}}\left(3-5\frac{{z}^{2}}{{r}^{2}}\right)+3\frac{{z}^{2}}{{r}^{2}}-1\right)\right)\right],\hfill \\ \hfill & {H}_{x}=\frac{z}{{r}^{5}}\left(3x\frac{l}{2m}-\frac{3{l}^{2}{\lambda }_{C}^{2}}{2{\bar{\rho }}_{0}^{2}}ty\right),\hfill \\ \hfill & {H}_{y}=\frac{z}{{r}^{5}}\left(3y\frac{l}{2m}-\frac{3{l}^{2}{\lambda }_{C}^{2}}{2{\bar{\rho }}_{0}^{2}}tx\right),\hfill \\ \hfill & {H}_{z}=\frac{l}{2m}\left(3\frac{{z}^{2}}{{r}^{2}}-1\right)\frac{1}{{r}^{3}}\hfill \end{align} \tag{ 83 }$$

We now transform these fields to the laboratory frame in which the particle moves along the z axis with a velocity ⟨u⟩ ≡ β according to the law

$$\begin{equation*}\langle z\rangle =\beta t\end{equation*}$$

Applying Lorentz transformations we get electric fields in the laboratory frame

$$\begin{align}\hfill & {E}_{x}^{\left(\text{lab}\right)}=\gamma \left({E}_{x}+\beta {H}_{y}\right),\hfill \\ \hfill & {E}_{y}^{\left(\text{lab}\right)}=\gamma \left({E}_{y}-\beta {H}_{x}\right),\hspace{25.0pt}{E}_{z}^{\left(\text{lab}\right)}={E}_{z}\hfill \end{align} \tag{ 84 }$$

Simultaneously, we need to transform the coordinates and the time as follows ⁸ :

$$\begin{align}\hfill \boldsymbol{\rho }=\left\{x,y\right\}=\text{inv},\enspace z\to \gamma \left(z-\beta t\right)= :{R}_{z},\quad t\to \gamma \left(t-\beta z\right)= :{T}_{z},\\ \hfill {r}^{2}\to \left({\rho }^{2}+{\gamma }^{2}{\left(z-\beta t\right)}^{2}\right)\enspace ,\quad \gamma ={\left(1-{\beta }^{2}\right)}^{-1/2}\end{align} \tag{ 85 }$$

We omit the magnetic fields, as to calculate the surface current below we need the electric field only.

Let us introduce a vector R = { ρ , γ(z − βt)} in the laboratory frame. The components of the electric field in this frame read

$$\begin{align}\hfill & {E}_{x}\left(\boldsymbol{r},t\right)=\gamma \frac{x}{{R}^{3}}\left(1+\frac{3}{4}\frac{{\bar{\rho }}_{0}^{2}}{{R}^{2}}\left(1-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)+\frac{1}{4}{\ell }^{2}{\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2}\left[3\frac{{T}_{z}^{2}}{{R}^{2}}\left(1-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)\right.\right.\hfill \\ \hfill & \left.\left.\qquad +3\frac{{R}_{z}^{2}}{{R}^{2}}-6\beta \frac{{R}_{z}{T}_{z}}{{R}^{2}}-1\right]\right)+\frac{\ell }{2m}3\beta \gamma y\frac{{R}_{z}}{{R}^{5}},\hfill \\ \hfill & {E}_{y}\left(\boldsymbol{r},t\right)=\gamma \frac{y}{{R}^{3}}\left(1+\frac{3}{4}\frac{{\bar{\rho }}_{0}^{2}}{{R}^{2}}\left(1-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)+\frac{1}{4}{\ell }^{2}{\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2}\left[3\frac{{T}_{z}^{2}}{{R}^{2}}\left(1-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)\right.\right.\hfill \\ \hfill & \left.\left.\qquad +3\frac{{R}_{z}^{2}}{{R}^{2}}-6\beta \frac{{R}_{z}{T}_{z}}{{R}^{2}}-1\right]\right)-\frac{\ell }{2m}3\beta \gamma x\frac{{R}_{z}}{{R}^{5}},\hfill \\ \hfill & {E}_{z}\left(\boldsymbol{r},t\right)=\frac{{R}_{z}}{{R}^{3}}\left(1+\frac{3}{4}\frac{{\bar{\rho }}_{0}^{2}}{{R}^{2}}\left(3-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)+\frac{1}{4}{\ell }^{2}{\left(\frac{{\lambda }_{c}}{{\bar{\rho }}_{0}}\right)}^{2}\left[3\frac{{T}_{z}^{2}}{{R}^{2}}\left(3-5\frac{{R}_{z}^{2}}{{R}^{2}}\right)\right.\right.\hfill \\ \hfill & \left.\left.\qquad +3\frac{{R}_{z}^{2}}{{R}^{2}}-1\right]\right).\hfill \end{align} \tag{ 86 }$$