Ultrarelativistic electrons in counterpropagating laser beams

For solving the equation of motion (1) the phases appearing in the fields arguments are expressed via the trajectory x(τ)

$$\begin{equation}{\phi }_{1}\left(\tau \right)\equiv {k}_{1}\cdot x\left(\tau \right)=\frac{{k}_{1}\cdot \bar{P}}{m}\tau +\delta {\phi }_{1}\left(\tau \right),\qquad {\phi }_{2}\left(\tau \right)\equiv {k}_{2}\cdot x\left(\tau \right)=\frac{{k}_{2}\cdot \bar{P}}{m}\tau +\delta {\phi }_{2}\left(\tau \right)\end{equation} \tag{ 5 }$$

where

$$\begin{equation}\delta {\phi }_{1}\equiv \int \mathrm{d}\tau \frac{{k}_{1}\cdot \delta P\left(\tau \right)}{m},\qquad \delta {\phi }_{2}\equiv \int \mathrm{d}\tau \frac{{k}_{2}\cdot \delta P\left(\tau \right)}{m},\end{equation} \tag{ 6 }$$

with $\delta {P}_{\mu }={P}_{\mu }\left(\tau \right)-{\bar{P}}_{\mu }$. The bar symbol designates time-averaged quantities. The key assumption lying in the basis of our derivation is

$$\begin{align}& \int \mathrm{sin}\enspace {\phi }_{1}\enspace \mathrm{d}\tau \approx -\frac{m}{{k}_{1}\cdot \bar{P}}\mathrm{cos}\enspace {\phi }_{1},\qquad \int \mathrm{sin}\enspace {\phi }_{2}\enspace \mathrm{d}\tau \approx -\frac{m}{{k}_{2}\cdot \bar{P}}\mathrm{cos}\enspace {\phi }_{2},\\ & \int \mathrm{sin}\left({\phi }_{1}-{\phi }_{2}\right)\mathrm{d}\tau \approx -\frac{m}{\left({k}_{1}-{k}_{2}\right)\cdot \bar{P}}\mathrm{cos}\left({\phi }_{1}-{\phi }_{2}\right),\end{align} \tag{ 7 }$$

as well as similar relations where in the right wing cos → sin and in the left wing sin → −cos. By employing this assumption, the four-momentum P of the particle can be derived. With the momentum, expressions for δϕ₁, δϕ₂ according to equation (6) are calculated under certain restrictions, which assure the validity of the assumption of equation (7).

Since the vector potential is independent on the transverse coordinates, the canonical momentum in these directions is conserved P_⊥(τ) = p_⊥ − eA(τ). Without loss of generality, we choose the initial transverse momentum p_⊥ to be on the x-axis. Then,

$$\begin{equation}{P}_{x}\left(\tau \right)={p}_{x}+m{\xi }_{1}\enspace \mathrm{cos}\enspace {\phi }_{1}+m{\xi }_{2}\enspace \mathrm{cos}\enspace {\phi }_{2},\end{equation} \tag{ 8 }$$

$$\begin{equation}{P}_{y}\left(\tau \right)=m{\xi }_{1}\enspace \mathrm{sin}\enspace {\phi }_{1}+m{\xi }_{2}\enspace \mathrm{sin}\enspace {\phi }_{2},\end{equation} \tag{ 9 }$$

where −ea_1,2 = mξ_1,2. Applying the assumption of equation (7), the x, y components of the trajectory read

$$\begin{equation}x\left(\tau \right)=\left(\frac{{p}_{x}}{m}\tau +\frac{m{\xi }_{1}}{{k}_{1}\cdot \bar{P}}\mathrm{sin}\enspace {\phi }_{1}+\frac{m{\xi }_{2}}{{k}_{2}\cdot \bar{P}}\mathrm{sin}\enspace {\phi }_{2}\right),\end{equation} \tag{ 10 }$$

$$\begin{equation}y\left(\tau \right)=-\left(\frac{m{\xi }_{1}}{{k}_{1}\cdot \bar{P}}\mathrm{cos}\enspace {\phi }_{1}+\frac{m{\xi }_{2}}{{k}_{2}\cdot \bar{P}}\mathrm{cos}\enspace {\phi }_{2}\right).\end{equation} \tag{ 11 }$$

Now let us consider the oscillations on the z axis

$$\begin{equation}\frac{\mathrm{d}{\mathbf{P}}_{z}}{\mathrm{d}\tau }=\frac{e}{m}{\mathbf{P}}_{\perp }{\times}\mathbf{B}.\end{equation} \tag{ 12 }$$

Employing equations (8), (9) and (4), one can find out that the terms scaling like ${\xi }_{1}^{2},{\xi }_{2}^{2}$ cancel. Therefore we have

$$\begin{equation}\frac{\mathrm{d}{\mathbf{P}}_{z}}{\mathrm{d}\tau }=-{p}_{x}\omega \left[{\xi }_{1}\enspace \mathrm{sin}\enspace {\phi }_{1}-{\xi }_{2}\enspace \mathrm{sin}\enspace {\phi }_{2}\right]-2\omega m{\xi }_{1}{\xi }_{2}\enspace \mathrm{sin}\left({\phi }_{1}-{\phi }_{2}\right).\end{equation} \tag{ 13 }$$

Accordingly, ${P}_{z}={\bar{P}}_{z}+\delta {P}_{z}$ with

$$\begin{equation}\delta {P}_{z}={p}_{x}\omega \left[\frac{m{\xi }_{1}}{{k}_{1}\cdot \bar{P}}\mathrm{cos}\enspace {\phi }_{1}-\frac{m{\xi }_{2}}{{k}_{2}\cdot \bar{P}}\mathrm{cos}\enspace {\phi }_{2}\right]+\frac{2{m}^{2}{\xi }_{1}{\xi }_{2}\omega }{\left({k}_{1}-{k}_{2}\right)\cdot \bar{P}}\mathrm{cos}\left({\phi }_{1}-{\phi }_{2}\right),\end{equation} \tag{ 14 }$$

where $\overline{P}$ is the time-averaged momentum, whose relation to the initial momentum of the electron before interacting with the laser pulses will be discussed in section 2.3. Integrating over τ, one obtains the z-component of the trajectory

$$\begin{equation}z\left(\tau \right)=\frac{{\bar{P}}_{z}}{m}\tau +\frac{2{m}^{2}{\xi }_{1}{\xi }_{2}\omega }{{\left[\left({k}_{1}-{k}_{2}\right)\cdot \bar{P}\right]}^{2}}\mathrm{sin}\left({\phi }_{1}-{\phi }_{2}\right)+{p}_{x}\omega \left[\frac{m{\xi }_{1}}{{\left({k}_{1}\cdot \bar{P}\right)}^{2}}\mathrm{sin}\enspace {\phi }_{1}-\frac{m{\xi }_{2}}{{\left({k}_{2}\cdot \bar{P}\right)}^{2}}\mathrm{sin}\enspace {\phi }_{2}\right].\end{equation} \tag{ 15 }$$

Let us now calculate the energy and its oscillatory part: $\varepsilon =\sqrt{{m}^{2}+{P}_{x}^{2}+{P}_{y}^{2}+{\left({\bar{P}}_{z}+\delta {P}_{z}\right)}^{2}}$. With equations (8) and (9),

$$\begin{align}\varepsilon & =\left[{m}^{2}+{m}^{2}{\xi }_{1}^{2}+{m}^{2}{\xi }_{2}^{2}+2{p}_{x}\left(m{\xi }_{1}\enspace \mathrm{cos}\enspace {\phi }_{1}+m{\xi }_{2}\enspace \mathrm{cos}\enspace {\phi }_{2}\right)\right.\\ & \quad {\left.+{\bar{P}}_{z}^{2}+{p}_{x}^{2}+2{\bar{P}}_{z}\delta {P}_{z}+2{m}^{2}{\xi }_{1}{\xi }_{2}\enspace \mathrm{cos}\left({\phi }_{1}-{\phi }_{2}\right)+\delta {P}_{z}^{2}\right]}^{1/2}.\end{align} \tag{ 16 }$$

Using δP_z given by equation (14) and recalling that $\left({k}_{1}-{k}_{2}\right)\cdot \bar{P}=-2\omega {\bar{P}}_{z}$, one can find out that the terms proportional to cos(ϕ₁ − ϕ₂) cancel each other. The expression for the energy may be further simplified to

$$\begin{equation}\varepsilon ={\left\{{m}^{2}+{m}^{2}{\xi }_{1}^{2}+{m}^{2}{\xi }_{2}^{2}+{p}_{x}^{2}+{\bar{P}}_{z}^{2}+\delta {P}_{z}^{2}+2{p}_{x}\left[\frac{m{\xi }_{1}}{\left(1-{\bar{v}}_{z}\right)}\mathrm{cos}\enspace {\phi }_{1}+\frac{m{\xi }_{2}}{\left(1+{\bar{v}}_{z}\right)}\mathrm{cos}\enspace {\phi }_{2}\right]\right\}}^{1/2}\end{equation} \tag{ 17 }$$

with ${k}_{1}\cdot \bar{P}=\omega \bar{\varepsilon }\left(1-{\bar{v}}_{z}\right)$, ${k}_{2}\cdot \bar{P}=\omega \bar{\varepsilon }\left(1+{\bar{v}}_{z}\right)$, where the average velocity on the z axis is defined as ${\bar{v}}_{z}={\bar{P}}_{z}/\bar{\varepsilon }$. With a Taylor expansion the following expression is obtained

$$\begin{equation}\varepsilon \approx \bar{\varepsilon }+\delta \varepsilon +\bar{\varepsilon }\enspace O{\left(\frac{\delta {P}_{z}}{\bar{\varepsilon }}\right)}^{2},\end{equation} \tag{ 18 }$$

where the average energy, effective mass, and the oscillatory part are defined as

$$\begin{equation}\bar{\varepsilon }=\sqrt{{m}_{{\ast}}^{2}+{p}_{x}^{2}+{\bar{P}}_{z}^{2}},\quad {m}_{{\ast}}\equiv m\sqrt{1+{\xi }_{1}^{2}+{\xi }_{2}^{2}},\quad \delta \varepsilon ={p}_{x}\omega \left[\frac{m{\xi }_{1}}{{k}_{1}\cdot \bar{P}}\mathrm{cos}\enspace {\phi }_{1}+\frac{m{\xi }_{2}}{{k}_{2}\cdot \bar{P}}\mathrm{cos}\enspace {\phi }_{2}\right].\end{equation} \tag{ 19 }$$

Notice that for vanishing transverse momentum p_x = 0 the energy is constant, in accordance with [51]. The expansion in equation (18) is justified if $\delta {P}_{z}\ll \bar{\varepsilon }$. Here we have taken into account that for an ultrarelativistic electron, the amplitude of δP_z is always larger than δɛ according to equations (14) and (19) and thus δɛ/ɛ < δP_z /ɛ ≪ 1. Taking into account the explicit form of δP_z , equation (14), the validity condition $\delta {P}_{z}\ll \bar{\varepsilon }$ reads

$$\begin{equation}\frac{m{\xi }_{1}{p}_{x}\omega }{{k}_{1}\cdot \bar{P}\bar{\varepsilon }}=\frac{{p}_{x}m{\xi }_{1}}{\left(1-{\bar{v}}_{z}\right){\bar{\varepsilon }}^{2}}\ll 1\end{equation} \tag{ 20 }$$

$$\begin{equation}\frac{m{\xi }_{2}{p}_{x}\omega }{{k}_{2}\cdot \bar{P}\bar{\varepsilon }}=\frac{{p}_{x}m{\xi }_{2}}{\left(1+{\bar{v}}_{z}\right){\bar{\varepsilon }}^{2}}\ll 1\end{equation} \tag{ 21 }$$

$$\begin{equation}\frac{2{m}^{2}{\xi }_{1}{\xi }_{2}\omega }{\left({k}_{1}-{k}_{2}\right)\cdot \bar{P}\bar{\varepsilon }}=\frac{{m}^{2}{\xi }_{1}{\xi }_{2}}{{\bar{v}}_{z}{\bar{\varepsilon }}^{2}}\ll 1.\end{equation} \tag{ 22 }$$

So far ϕ₁, ϕ₂ were not specified yet. With the help of δɛ, δP_z we evaluate δϕ₁, δϕ₂ and thus obtain the phases ϕ₁, ϕ₂. Accordingly, the validity criterion for the basic assumption of this derivation, equation (7), is determined. Substituting equations (14) and (19) in (6) we have

$$\begin{equation}\delta {\phi }_{1}={{\Phi}}_{1}+{C}_{1}\enspace \mathrm{sin}\enspace {\phi }_{2}-{C}_{12}\enspace \mathrm{sin}\left({\phi }_{1}-{\phi }_{2}\right),\qquad \delta {\phi }_{2}={{\Phi}}_{2}+{C}_{2}\enspace \mathrm{sin}\enspace {\phi }_{1}+{C}_{12}\enspace \mathrm{sin}\left({\phi }_{1}-{\phi }_{2}\right),\end{equation} \tag{ 23 }$$

where Φ₁, Φ₂ are arbitrary constants and the coefficients are

$$\begin{equation}{C}_{1}\equiv \frac{2{p}_{x}m{\xi }_{2}{\omega }^{2}}{{\left({k}_{2}\cdot \bar{P}\right)}^{2}}=\frac{2{p}_{x}m{\xi }_{2}}{{\varepsilon }^{2}{\left(1+{\bar{v}}_{z}\right)}^{2}}\qquad {C}_{2}\equiv \frac{2{p}_{x}m{\xi }_{1}{\omega }^{2}}{{\left({k}_{1}\cdot \bar{P}\right)}^{2}}=\frac{2{p}_{x}m{\xi }_{1}}{{\varepsilon }^{2}{\left(1-{\bar{v}}_{z}\right)}^{2}}\qquad {C}_{12}\equiv \frac{2{m}^{2}{\xi }_{1}{\xi }_{2}{\omega }^{2}}{{\left[\left({k}_{1}-{k}_{2}\right)\cdot \bar{P}\right]}^{2}}=\frac{{m}^{2}{\xi }_{1}{\xi }_{2}}{2{\varepsilon }^{2}{\bar{v}}_{z}^{2}}.\end{equation} \tag{ 24 }$$

Equation (5) together with equation (23) form an implicit system for the solution of the phases. Without loss of generality, we assumed that ${\bar{v}}_{z}{ >}0$, i.e. the particle copropagates with the ξ₁ beam, leading to asymmetry between the two beams. As a consequence, ${k}_{1}\cdot \bar{P}\ll {k}_{2}\cdot \bar{P}$, so that if the beams amplitudes are of the same order of magnitude, C₂ is considerably larger than C₁. In the following we assume that C₁, C₁₂ ≪ 1, yielding the following expressions

$$\begin{equation}{\phi }_{1}\left(\tau \right)\approx {{\Phi}}_{1}+\frac{{k}_{1}\cdot \bar{P}}{m}\tau ,\qquad {\phi }_{2}\left(\tau \right)\approx {{\Phi}}_{2}+\frac{{k}_{2}\cdot \bar{P}}{m}\tau +\frac{2{p}_{x}m{\xi }_{1}{\omega }^{2}}{{\left({k}_{1}\cdot \bar{P}\right)}^{2}}\mathrm{sin}\enspace {\phi }_{1}.\end{equation} \tag{ 25 }$$

In order to prove the consistency of this conclusion, one should accomplish two things. First, one has to show that the contributions of C₁, C₁₂ to the momentum are of second order, justifying the neglection. For this purpose, we consider a general function F with the following argument ϕ(τ) = ϕ₀(τ) + ν sin f(τ), where ν is a small constant and ϕ₀(τ), f(τ) are general functions. Taylor expanding with respect to ν yields

$$\begin{equation}F\left[\phi \left(\tau \right)\right]\approx F\left[{\phi }_{0}\left(\tau \right)\right]+\nu {F}^{\prime }\left[{\phi }_{0}\left(\tau \right)\right]\mathrm{sin}\enspace f\left(\tau \right).\end{equation} \tag{ 26 }$$

In our case, ϕ₀ designates the approximated phases ϕ₁ or ϕ₂ given in equation (25) and ν is either C₁ or C₁₂, ϕ stands for the full phases including the neglected terms proportional to C₁₂, C₁, and F(ϕ) either  cos(ϕ) or  sin ϕ, where stands for the amplitudes of the various momentum oscillations appearing in equations (8), (9) and (14). Since F' ∼ , the correction scales as O(ν). One should notice that the amplitude of the momentum oscillations are assumed to be considerably smaller with respect to the particle energy, being the dominant energy scale. Hence, is a small parameter and the corrections corresponding to C₁, C₁₂ may be neglected, up to the second order.

Second, one should verify that the approximation of equation (7) indeed holds. Plugging the phases equation (25) into equation (7) we notice that all the three integrals take the form $\mathcal{I}\equiv \int \mathrm{cos}\left[\alpha \tau +\beta \right.$ $\left.\mathrm{sin}\left(\kappa \tau \right)\right]\mathrm{d}\tau$ with different choices of α, β, κ. In order to calculate this integral, we recall the identity

$$\begin{equation}{\text{e}}^{\text{i}\beta \mathrm{sin}\left(\kappa \tau \right)}=\sum\limits _{s}{J}_{s}\left(\beta \right){\mathrm{e}}^{\mathrm{i}s\kappa \tau },\end{equation} \tag{ 27 }$$

where J_s (β) is the Bessel function. Multiplying e^iατ on both sides, one readily obtains the real and imaginary part, respectively, as

$$\begin{equation}\mathrm{cos}\left[\alpha \tau +\beta \enspace \mathrm{sin}\left(\kappa \tau \right)\right]=\sum\limits _{s}{J}_{s}\left(\beta \right)\mathrm{cos}\left[\left(\alpha +s\kappa \right)\tau \right];\mathrm{sin}\left[\alpha \tau +\beta \enspace \mathrm{sin}\left(\kappa \tau \right)\right]=\sum\limits _{s}{J}_{s}\left(\beta \right)\mathrm{sin}\left[\left(\alpha +s\kappa \right)\tau \right].\end{equation} \tag{ 28 }$$

The integral can thus be obtained as

$$\begin{equation}\mathcal{I}=-\frac{1}{2}\sum\limits _{s}{J}_{s}\left(\beta \right)\frac{1}{\alpha +s\kappa }\left[{\text{e}}^{\text{i}\left(\alpha +s\kappa \right)\tau }+{\text{e}}^{-\text{i}\left(\alpha +s\kappa \right)\tau }\right].\end{equation} \tag{ 29 }$$

For a certain β we know that J_s (β) vanishes if the index s is larger enough than β. Therefore, further simplification can be accomplished if

$$\begin{equation}s\kappa /\alpha \lesssim \beta \kappa /\alpha \ll 1.\end{equation} \tag{ 30 }$$

The integral is thus approximated by

$$\begin{equation}\mathcal{I}\approx -\frac{1}{\alpha }\mathrm{cos}\left[\alpha \tau +\beta \enspace \mathrm{sin}\left(\kappa \tau \right)\right],\end{equation} \tag{ 31 }$$

where equation (28) has been considered. This result is in agreement with equation (7). Now let us find the conditions for which equation (30) is satisfied for all three cases. For the first integral, β vanishes and equation (30) is trivially fulfilled. For the second case, one has $\alpha =\left({k}_{2}\cdot \bar{P}\right)/{m}^{2},\kappa =\left({k}_{1}\cdot \bar{P}\right)/{m}^{2}$ and β = C₂, so that equation (30) yields

$$\begin{equation}\frac{{p}_{x}m{\xi }_{1}}{{\bar{\varepsilon }}^{2}}\ll \frac{\left(1+{\bar{v}}_{z}\right)\left(1+{\bar{v}}_{z}\right)}{2}.\end{equation} \tag{ 32 }$$

For the third integral β, κ are as in the second case but $\alpha =\left[\left({k}_{2}-{k}_{1}\right)\cdot \bar{P}\right]/{m}^{2}$, imposing the condition

$$\begin{equation}\frac{{p}_{x}m{\xi }_{1}}{{m}_{{\ast}}^{2}}\ll {\bar{v}}_{z}\left(1-{\bar{v}}_{z}\right).\end{equation} \tag{ 33 }$$

Thus, equation (7) was explicitly shown to be valid, given that equations (32) and (33) are satisfied. Combining C₁, C₁₂ ≪ 1 with equations (20)–(22), (32) and (33), yields the final validity criteria

$$\begin{equation}\frac{{m}^{2}{\xi }_{1}{\xi }_{2}}{{\varepsilon }^{2}}\ll {\bar{v}}_{z}\enspace \mathrm{min}\left[1,2{\bar{v}}_{z}\right],\qquad \frac{{p}_{x}m{\xi }_{2}}{{\varepsilon }^{2}}\ll {\left(1+{\bar{v}}_{z}\right)}^{2},\qquad \frac{{p}_{x}m{\xi }_{1}}{{\varepsilon }^{2}}\ll {\bar{v}}_{z}\left(1-{\bar{v}}_{z}\right).\end{equation} \tag{ 34 }$$

Let us conclude the derivation. The final expressions for the trajectory and momentum are equations (10), (11), and (15) and equations (8), (9), (14), (19), correspondingly. The phases ϕ₁(τ) and ϕ₂(τ) are given by equation (25). The validity criteria corresponding to this solution are equation (34). In the ultrarelativistic regime $1-{\bar{v}}_{z}\ll 1$ they are simplified to

$$\begin{equation}\frac{{p}_{x}m{\xi }_{2}}{2{\varepsilon }^{2}},\frac{{m}^{2}{\xi }_{1}{\xi }_{2}}{{\varepsilon }^{2}},\frac{2{p}_{x}m{\xi }_{1}}{{m}_{{\ast}}^{2}}\ll 1.\end{equation} \tag{ 35 }$$

The above criteria can be fulfilled in a scenario where an ultrarelativistic electron moves along the laser propagation direction with a small deviating angle. Alternatively, one may write the instantaneous momentum in a covariant form as follows

$$\begin{align}{P}_{\mu }\left(\tau \right)& ={\bar{P}}_{\mu }-e\left[{A}_{1}^{\mu }\left({\phi }_{1}\right)+{A}_{2}^{\mu }\left({\phi }_{2}\right)\right]+{k}_{1}^{\mu }\left[\frac{ep\cdot {A}_{1}\left({\phi }_{1}\right)}{{k}_{1}\cdot \bar{P}}-\frac{{A}_{1}\left({\phi }_{2}\right)\cdot {A}_{2}\left({\phi }_{2}\right)}{\left({k}_{1}-{k}_{2}\right)\cdot \bar{P}}\right]\\ & \quad +{k}_{2}^{\mu }\left[\frac{ep\cdot {A}_{2}\left({\phi }_{2}\right)}{{k}_{2}\cdot \bar{P}}+\frac{{A}_{1}\left({\phi }_{2}\right)\cdot {A}_{2}\left({\phi }_{2}\right)}{\left({k}_{1}-{k}_{2}\right)\cdot \bar{P}}\right].\end{align} \tag{ 36 }$$

One may verify that in the case if one of the laser beams vanishes, our result equation (36) recovers the familiar plane wave solution [5].

The above derivation expresses the physical quantities of interest, namely the trajectory and the four-momentum, as a function of the proper time τ. However, for practical applications it is favorable to use the laboratory time as the independent variable. The two quantities are simply related through $\mathrm{d}t=\frac{\varepsilon }{m}\mathrm{d}\tau$. Performing the integration we obtain

$$\begin{equation}t\left(\tau \right)=\frac{\bar{\varepsilon }}{m}\tau +{p}_{x}\omega \left[\frac{m{\xi }_{1}}{{\left({k}_{1}\cdot \bar{P}\right)}^{2}}\enspace \mathrm{sin}\enspace {\phi }_{1}\left(\tau \right)+\frac{m{\xi }_{2}}{{\left({k}_{2}\cdot \bar{P}\right)}^{2}}\enspace \mathrm{sin}\enspace {\phi }_{2}\left(\tau \right)\right].\end{equation} \tag{ 37 }$$

The latter along with x(τ) provides a parametric description of the particle coordinate as a function of the laboratory time. Alternatively, one may further approximate the phases. We start by writing equation (37) as

$$\begin{equation}\tau =\frac{m}{\bar{\varepsilon }}\left\{t-{p}_{x}\omega \left[\frac{m{\xi }_{1}}{{\left({k}_{1}\cdot \bar{P}\right)}^{2}}\mathrm{sin}\enspace {\phi }_{1}\left(\tau \right)+\frac{m{\xi }_{2}}{{\left({k}_{2}\cdot \bar{P}\right)}^{2}}\mathrm{sin}\enspace {\phi }_{2}\left(\tau \right)\right]\right\}.\end{equation} \tag{ 38 }$$

Substituting (38) into the phase ϕ₁ given in (25) one obtains

$$\begin{equation}{\phi }_{1}={{\Phi}}_{1}+{\omega }_{1}t-{p}_{x}\omega {\omega }_{1}\left[\frac{m{\xi }_{1}}{{\left({k}_{1}\cdot \bar{P}\right)}^{2}}\mathrm{sin}\enspace {\phi }_{1}+\frac{m{\xi }_{2}}{{\left({k}_{2}\cdot \bar{P}\right)}^{2}}\mathrm{sin}\enspace {\phi }_{2}\right],\end{equation} \tag{ 39 }$$

where ${\omega }_{1}\equiv \left(1-{\bar{v}}_{z}\right)\omega$. This equation is implicit, since ϕ₁ appears in both sides. Nevertheless, it proves useful as a starting point for approximation of the phases, as we immediately show. According to the validity condition equation (34), one notices that the coefficients of the sine functions in equation (39) are much smaller than 1. As a result, equation (26) may be employed here. The fact that equation (39) is implicit (ϕ₁ appears in both sides) poses no difficulty, since the argument f(τ) in (26) is general and has no influence on the final result. Due to (26) and according to the same reasoning that led us to neglect C₁, C₁₂, they may be omitted, leading to

$$\begin{equation}{\phi }_{1}\left(t\right)\approx {{\Phi}}_{1}+{\omega }_{1}t,\qquad {\phi }_{2}\left(t\right)\approx {{\Phi}}_{2}+{\omega }_{2}t+\frac{2{p}_{x}m{\xi }_{1}{\omega }^{2}}{{\left({k}_{1}\cdot \bar{P}\right)}^{2}}\mathrm{sin}\left({\omega }_{1}t\right),\end{equation} \tag{ 40 }$$

where ${\omega }_{2}\equiv \left(1+{\bar{v}}_{z}\right)\omega$. Hence, one observes that ω₁, ω₂ are the characteristic oscillation frequencies associated with the ξ₁, ξ₂ beams, respectively. Notice that according to our convention the particle copropagates with the ξ₁ beam, so that ${\bar{v}}_{z}$ is positive, and hence ω₂ is considerably larger than ω₁, which indicates the non-resonant regime of interaction.

With the obtained analytical expression for the electron momentum and coordinate, we study in this section the main characteristics of the motion. As the dynamics is strongly effected by both of the laser beams, we would expect to find some unusual features in the electron trajectory, where the acceleration is large and which may yield radiation emission deviating from the LCFA results based on the Baier–Katkov technique [49]. We inspect the electron velocity in all components for three different field parameters in figure 1, featuring various behaviors. The results shown in the figure are obtained within the analytical treatment presented above and proved by the fully numerical solutions of equation (1). In all cases the initial transverse momentum vanishes p_x = 0 and the energy is ɛ = 2.6m_*, corresponding to ω₂/ω₁ $\approx$ 25. The plots present a time interval of 2π/ω₁, so it consists of one cycle of the ξ₁ beam and about 25 cycles of the ξ₂ beam.

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In panel (a) the laser parameters are ξ₁ = 50, ξ₂ = 1. In the x–y plane the particle performs a cyclic motion with a radius of mξ₁/ɛ and a frequency ω₁ and on top of it rapid oscillations with frequency ω₂ and amplitude mξ₂/ɛ. According to equation (14), the amplitude of the oscillation on the z axis scales as ∼m² ξ₁ ξ₂/ɛ² and is, therefore, considerably smaller as compared to those in the x, y axes.

Panel (b) depicts the case of ξ₁ = ξ₂ = 20. In the x–y plane the oscillations amplitude are now identical, so that the particle moves in circles with frequency ω₂ according to ξ₂. An interesting point is that the origin of the circle also exhibits a cyclic motion due to ξ₁ with a frequency of ω₁. Both the fast ξ₂ circle and the slow ξ₁ circle have the same radius because of the identical oscillation amplitudes. In addition, one can observe that the tilting angle of the total velocity with respect to the z axis is gradually changing. The reason is that the oscillation frequency on the z axis is ω₂ − ω₁. As a result, the relative phase between v_z and v_x for example gradually increases during the time interval under consideration from 0 to 2π.

Panel (c) presents the dynamics for ξ₁ = 1, ξ₂ = 50. It is quite similar to the previous case, but now the radius of the slow ξ₁ circle is negligible, such that the motion takes the form of a single circle with time dependent tilt.

With respect to radiation emission, the more irregular the trajectory is, the more interesting is the spectral shape. Hence, in the following we concentrate on the ξ₁ ≫ ξ₂ case, like in panel (a) of figure 1, where the dynamics is much more complex. Figure 2 shows a two dimensional projection of the velocity on the x–y plane for ξ₁ = 50, ξ₂ = 1 with three different particle energies ɛ.

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Panel (a) corresponds to ɛ = 130m. As mentioned in figure 1, one can see that the dynamics is a combination of a large circle due to ξ₁ and rapid oscillations corresponding to ξ₂, which have a smooth sine-shape, see in the inset. When the energy is increased, see panel (b) with ɛ = 182m, several interesting changes take place. First of all, the number of the ξ₂ oscillations contained in one cycle of ξ₁ increases since the ratio of the frequencies, ω₂/ω₁, is now about 51 instead of 25 in panel (a). Furthermore, the radius of the circle as well as the amplitude of the small oscillations becomes smaller. This is because the amplitude of the transverse velocity v_⊥ ∼ mξ₁/ɛ decreases with the energy increase. More interestingly, a sharp spike-like feature emerges for each cycle of ξ₂ oscillation. It should be emphasized that the time scale corresponding to these spike-like features is significantly shorter than both 1/ω₁ and 1/ω₂.

In order to shed light on this spike-like feature, we take advantage of the approximated phases equation (40) and derive from the y component of the trajectory (11) the corresponding acceleration

$$\begin{equation}{\dot {v}}_{y}=\frac{m{\xi }_{1}{\omega }_{1}}{\bar{\varepsilon }}\mathrm{cos}\enspace {\phi }_{1}+\frac{m{\xi }_{2}{\omega }_{2}}{\bar{\varepsilon }}\mathrm{cos}\enspace {\phi }_{2}=\frac{{m}^{2}}{{\bar{\varepsilon }}^{2}}\left({\chi }_{1}\enspace \mathrm{cos}\enspace {\phi }_{1}+{\chi }_{2}\enspace \mathrm{cos}\enspace {\phi }_{2}\right),\end{equation} \tag{ 41 }$$

where in the last expression we take into account that the quantum parameter is proportional to the acceleration $\chi ={\varepsilon }^{2}\vert \dot {v}\vert /{m}^{3}$ [49]. Please note that here $\dot {a}$ refers to the derivative of time t. Here χ₁ = ξ₁ ɛω₁/m, and χ₂ = ξ₂ ɛω₂/m are the quantum parameters induced, respectively, by beams 1 and 2. Let us take a close look at the time interval corresponding to 0 < ϕ₁ < π/2. One can see that as long as χ₂ < χ₁, the acceleration does not change its sign. Namely, the velocity will monotonously decrease, as the case in figure 2(a). Increasing the energy results in higher values of the ratio ω₂/ω₁, and at a certain point χ₂ exceeds χ₁. When χ₂ becomes large enough, the acceleration ${\dot {v}}_{y}$ will change its sign during the time interval. If χ₂ is only slightly higher than χ₁, the acceleration is positive for a very short time, leading to sharp spikes, as encountered in figure 2(b). In case χ₂ is significantly larger than χ₁, the acceleration is positive about half of the time, giving rise to the whirl appearing in figure 2(c), where the energy is further increased to ɛ = 250m. The impact of these phenomena on the radiation emission has been explored in reference [41].

Finally, let us examine the influence of the transverse momentum p_x . From the final expressions for the momentum and energy, one observes that this quantity has several contributions. First, it gives rise to the oscillations in the longitudinal momentum P_z and the energy ɛ, see equations (14) and (19), respectively. This means that the energy of the electron in the field is not constant anymore. Moreover, the non-zero transverse momentum also adds a slow sine-term (with frequency ω₁) to the phase ϕ₂, see equation (25). As a result, the rapid oscillations corresponding to ξ₂ are periodically modulated. This phenomenon is demonstrated in figure 3, where the y-component of velocity is plotted as a function of time within half a cycle of ξ₁. To verify our analytical results (black), the numerical solution is also shown in this figure as a blue line. The agreement between the numerical solution and the analytical one is excellent, as the two curves are on top of each other. In addition, as we expected, the frequency of the small oscillations increases with time up to t/T ≈ 7, with T = 2π/ω, and then gradually decreases again.

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From the discussion above, we can see that the drift momentum of the particle in the laser fields, especially the average energy in the field, is an essential parameter for our approximation. However, the drift momentum depends on the asymptotic momentum of the particle before entering in the laser fields as well as the way of switching on the laser pulses. In this section, we will derive the relation explicitly. The relation between ${\bar{P}}_{\mu }$ and the asymptotic momentum of the particle p_μ is governed by the ponderomotive force [62, 63], arising from the turn on process of the laser fields:

$$\begin{equation}\frac{\mathrm{d}{\bar{P}}_{z}}{\mathrm{d}\tau }=-\frac{1}{2m}\frac{\partial }{\partial z}\bar{{\left\vert e\mathbf{A}\right\vert }^{2}}.\end{equation} \tag{ 42 }$$

Substituting equation (2) and keeping the envelope functions g₁, g₂, we have

$$\begin{equation}\frac{\mathrm{d}{\bar{P}}_{z}}{\mathrm{d}\tau }=\frac{\omega m}{2}\left[{\xi }_{1}^{2}\frac{\mathrm{d}}{\mathrm{d}{\phi }_{1}}{g}_{1}^{2}\left[{\phi }_{1}\left(\tau \right)\right]-{\xi }_{2}^{2}\frac{\mathrm{d}}{\mathrm{d}{\phi }_{2}}{g}_{2}^{2}\left[{\phi }_{2}\left(\tau \right)\right]\right].\end{equation} \tag{ 43 }$$

Suppose that the copropagating laser pulse with the amplitude ξ₁ is turned on first. During this process the second integral for ${\bar{P}}_{z}$ is vanishing, and the particle momentum reads

$$\begin{equation}{\bar{P}}_{z}^{\left(1\right)}={p}_{z}+\frac{\omega m{\xi }_{1}^{2}}{2}{\int }^{\tau }\mathrm{d}{\tau }^{\prime }\frac{\mathrm{d}}{\mathrm{d}{\phi }_{1}}{g}_{1}^{2}\left[{\phi }_{1}\left({\tau }^{\prime }\right)\right]={p}_{z}+\frac{{m}^{2}{\xi }_{1}^{2}\omega }{2\left({k}_{1}\cdot p\right)}.\end{equation} \tag{ 44 }$$

Here ϕ₁ = (k ⋅ p)τ/m, because in the absence of the counterpropagating pulse k₁ ⋅ P is exactly conserved. It is worthwhile to mention that this result is similar to the one corresponding to the plane wave case. Now the second pulse is turned on. Its contribution to the momentum is given by

$$\begin{equation}{\bar{P}}_{z}={\bar{P}}_{z}^{\left(1\right)}-\frac{\omega m{\xi }_{2}^{2}}{2}{\int }^{\tau }\mathrm{d}{\tau }^{\prime }\frac{\mathrm{d}}{\mathrm{d}{\phi }_{2}}{g}_{2}^{2}\left[{\phi }_{2}\left({\tau }^{\prime }\right)\right].\end{equation} \tag{ 45 }$$

Recalling the approximation derived above equation (26), and assuming that the pulse is turned on adiabatically, namely g₂'/g₂ → 0, the oscillatory part of the phase may be omitted, yielding ${g}_{2}\left({\phi }_{2}\right)\approx {g}_{2}$ $\left(\frac{{k}_{2}\cdot \bar{P}}{m}\tau \right)$. Since the first pulse effect comes into play through the neglected oscillatory term in ϕ₂, it does not influence the integration. We further assume that ${k}_{2}\cdot \bar{P}$ remains constant during the turn on of the second pulse. Then, the integral in (45) is straightforwardly carried out, yielding for ${\bar{P}}_{z}$ and $\bar{\varepsilon }$:

$$\begin{equation}{\bar{P}}_{z}={p}_{z}+\frac{1}{2}\left[\frac{{m}^{2}{\xi }_{1}^{2}}{{k}_{1}\cdot p}-\frac{{m}^{2}{\xi }_{2}^{2}}{{k}_{2}\cdot {\bar{P}}^{\left(1\right)}}\right]\omega ,\qquad \bar{\varepsilon }={p}_{0}+\frac{1}{2}\left[\frac{{m}^{2}{\xi }_{1}^{2}}{{k}_{1}\cdot p}+\frac{{m}^{2}{\xi }_{2}^{2}}{{k}_{2}\cdot {\bar{P}}^{\left(1\right)}}\right]\omega ,\end{equation} \tag{ 46 }$$

where equation (44), and $\bar{\varepsilon }=\sqrt{{m}_{{\ast}}^{2}+{\bar{P}}_{z}^{2}}$ were employed. Hence

$$\begin{equation}{\bar{P}}^{\mu }={p}^{\mu }+\frac{{m}^{2}{\xi }_{1}^{2}}{2\left({k}_{1}\cdot p\right)}{k}_{1}^{\mu }+\frac{{m}^{2}{\xi }_{2}^{2}}{2\left[{k}_{2}\cdot {\bar{P}}^{\left(1\right)}\right]}{k}_{2}^{\mu }.\end{equation} \tag{ 47 }$$

Examining the final momentum (47), one may observe that our assumption ${k}_{2}\cdot \bar{P}={k}_{2}\cdot {\bar{P}}^{\left(1\right)}$ was justified. We underline that

$$\begin{equation}{k}_{1}\cdot \bar{P}={k}_{1}\cdot p+\frac{{m}^{2}{\xi }_{2}^{2}\left({k}_{1}\cdot {k}_{2}\right)}{2\left[{k}_{2}\cdot {\bar{P}}^{\left(1\right)}\right]},\qquad {k}_{2}\cdot \bar{P}={k}_{2}\cdot p+\frac{{m}^{2}{\xi }_{1}^{2}\left({k}_{1}\cdot {k}_{2}\right)}{2\left[{k}_{1}\cdot p\right]}.\end{equation} \tag{ 48 }$$

Namely, neither k₁ ⋅ P nor k₂ ⋅ P are conserved. One may observe that k₁ ⋅ P is modified during the rise of the counterpropagating pulse and vice versa. In case the counterpropagating beam is turned on first, an analogous derivation leads to

$$\begin{equation}{\bar{P}}^{\mu }={p}^{\mu }+\frac{{m}^{2}{\xi }_{1}^{2}}{2\left[{k}_{1}\cdot {\bar{P}}^{\left(2\right)}\right]}{k}_{1}^{\mu }+\frac{{m}^{2}{\xi }_{2}^{2}}{2\left({k}_{2}\cdot p\right)}{k}_{2}^{\mu },\end{equation} \tag{ 49 }$$

where ${\bar{P}}_{\mu }^{\left(2\right)}={p}_{\mu }+\frac{{m}^{2}{\xi }_{1}^{2}}{2\left({k}_{2}\cdot p\right)}{k}_{2\mu }$.

The relation between the drift momentum and the asymptotic initial momentum has been also investigated by numerically solving the Lorentz equation (1) and comparing with the analytical results. Table 1 presents the average four-momentum of the electron after both laser beams are turned-on, corresponding to different initial momenta and intensities of the lasers. Since the order by which the lasers are turned-on affects the final state, the table contains both options. For the sake of simplicity, we assume the initial p_⊥ = 0 for all situations. From the expression for the final momentum equations (47) and (49), one can see that two factors determine which of the beams will be dominant. The obvious one is the corresponding field intensity. The surprising one is the relative direction between the propagation direction of the particle and the beam under consideration. It stems from the denominator $k\cdot \bar{P}$, namely, counterpropagating beams have lower influence than copropagating beams.

Table 1. The average four-momentum of an electron after both laser beams are turned-on. We consider three different cases for different initial momentum p_z and field parameters. In all the cases, the initial transverse momentum is chosen to be zero such that ${\bar{P}}_{x}={\bar{P}}_{y}=0$. The fifth column, named order, indicates the order by which the two laser beams are turned-on; (a) the ξ₁ beam is turned-on first. (b) The ξ₂ beam is turned-on first. The four-momentum is given in units of the electron rest mass m. The superscript N designates the numerical calculation and A the analytical one.

Case	ξ1	ξ2	pz	Order	${\left(\bar{\varepsilon },{\bar{P}}_{z}\right)}^{\mathrm{A}}$	${\left(\bar{\varepsilon },{\bar{P}}_{z}\right)}^{\mathrm{N}}$
1	10	10	0	a	(51.495, 49.505)	(51.502, 49.450)
				b	(51.495, −49.505)	(51.502, −49.450)
2	3	20	20	a	(200.637, 199.613)	(200.637, 199.612)
				b	(25.471, 15.452)	(25.835, 15.321)
3	30	2	−1	a	(187.816, 185.391)	(187.815, 185.390)
				b	(43.522, 31.451)	(29.531, −0.396)

In the first case the two beams have identical intensity and the particle is initially at rest, so the only thing that breaks the symmetry is the turn-on order. It demonstrates that a given beam will have a stronger influence if it is the first to be turned on. The reason is that after the turned-on, the particle will copropagate with the first beam and thus this beam will have a large influence in the final results. This also reflects in the direction of the average momentum as in this case the particle always copropagates with the first beam at the end, see in table 1.

For the second case appearing in the table, one may naively assume the ξ₂ beams should be dominant, since ξ₂/ξ₁ ∼ 7 and the contribution to the final momentum of each beam scales like ∼ξ². However, due to the fact that ξ₁ is copropagating, its effect is actually of the same order of magnitude as of the ξ₂ beam. This can be seen by the fact that the order of the turn-on causes an order of magnitude difference between the final energies. Namely, when ξ₁ is turned-on first (order (a)), the particle is first accelerated to ultrarelativistic energy and then slightly decelerated when ξ₂ is turned-on. The final energy is about ɛ ≈ 200m, which is much larger than the final energy ɛ ≈ 25m of the second scenario (order (b)), where the particle is first decelerated and then accelerated.

In the third case a new situation is encountered. The particle flips its direction of motion during the turn-on of the second beam if the ξ₂ beam is turn-on first, order (b) in the table. One may see that it initially propagates to the left and only after the second pulse rises it flips direction and propagates to the right. Both from analytical and from experimental perspectives, such a scenario should be avoided. From an experimental point of view, as it will lead to collisions of electrons in the beam with those following them. From analytical perspective, since a direction flip implies that the particle average velocity should vanish at a certain point in the middle of the turn-on process, violating the validity conditions. Indeed, the analytical expression in this case fails to reproduce the numerical result.

To complete the discussion, we specify several considerations which were taken into account when choosing the above parameters. First, we made sure that the validity criteria derived above are met. Second, the final propagation direction is always copropagating with the first turn-on beam, in agreement with the convention introduced in the previous subsection. Third, both laser amplitudes were chosen to be higher than 1. Since the contribution of each beam scales as ∝ξ², the influence of a beam with nonrelativistic intensity on the final momentum can be neglected.

In the following, the accuracy of the analytical solution derived in the previous sections is systematically put to a test. For the sake of this purpose, we define the relative deviation of the analytical prediction (subscript a) of a quantity X with respect to the numerically calculated value (subscript n) as follows

$$\begin{equation}{{\Delta}}_{X}\equiv \frac{1}{2T}{\int }_{-T}^{T}\mathrm{d}t\left\vert \frac{{X}_{\mathrm{a}}-{X}_{\mathrm{n}}}{{X}_{\mathrm{n}}}\right\vert ,\end{equation} \tag{ 50 }$$

with X being either the transverse velocity ${v}_{\perp }\equiv \sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$, the longitudinal one v_z or the energy ɛ. The integration time is taken to infinity, i.e. T → ∞. These deviations are explored as a function of ${{\epsilon}}_{1}\equiv 2{p}_{x}m{\xi }_{1}/{m}_{{\ast}}^{2}$ and ${{\epsilon}}_{2}\equiv {m}^{2}{\xi }_{1}{\xi }_{2}/{\bar{\varepsilon }}^{2}$, being the small parameters of the derivation for ultrarelativistic particles, see equation (35). In the following we restrict ourselves to ξ₁ > ξ₂, and hence the third small parameter is by definition smaller than ₂, namely ${p}_{x}m{\xi }_{2}/\left(2{\bar{\varepsilon }}^{2}\right){< }{{\epsilon}}_{2}$. For quantitative comparison of the analytical and numerical quantities, we specify the arbitrary constants Φ₁, Φ₂ in the phases ϕ₁, ϕ₂, which is accomplished in two ways, see figure 4. The first way (blue solid line) is to write down Φ₁ = k₁ ⋅ x₀, Φ₂ = k₂ ⋅ x₀ where x₀ denotes the temporal and spatial location of the particle at the moment when the turn-on process is over, associated with the numerical calculation. The second way (blue dashed line) is to advance the analytical solution in time from t → −∞ using the relation (47). It should be mentioned that this approach is not fully analytical since ${k}_{1}\cdot \bar{P}$ is not constant during the turn-on of the second pulse, but rather depends on ξ₁, which in its turn depends on ${k}_{1}\cdot \bar{P}$ through the phase ϕ₁. Nevertheless, assuming that the turn-on process is adiabatic, ξ₁ barely changes from one time step to its subsequent one, so that one can use the value of the field amplitude from the previous time step.

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Before our broader parameter survey, it is worthwhile to take a close look at a specific case in order to gain intuition regarding the nature of deviation. Figure 4 presents the longitudinal velocity for ξ₁ = 50, ɛ = 200m, p_x = 0 for a different ξ₂ value: ξ₂ = 20 and ξ₂ = 40 corresponding to ₁ = 0.05 (upper plot) and ₁ = 0.1 (lower plot), respectively. One can see that the main deviation stems from an inaccuracy in the phases ϕ₁, ϕ₂, rather than in the amplitudes. Moreover, the comparison between the dashed and solid curves in figure 4 demonstrates that the two approaches to determine the phases Φ₁, Φ₂ give similar results and the first method was employed in the following calculations.

Figure 5 depicts the relative deviations Δ_⊥, Δ_z , Δ defined in equation (50) as a function of ₁, ₂. We fixed the parameters ξ₁ = 100, ɛ = 200m and varied ξ₂, p_x in the ranges 0 < ξ₂ < 60, 0 < p_x < 15m, respectively. The initial momentum on the z axis was tuned in order to keep the energy ɛ constant. One may notice that for vanishing ξ₂, the analytical calculation is accurate regardless of the value of p_x . This occurs due to the fact that vanishing of ξ₂ corresponds to the plane wave limit, where the analytical solution equation (36) is exact without any restriction.

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Furthermore, we can see from all three panels in figure 5 that the influence of the small parameter ₂ is considerably stronger as compared to that of ₁. This is because ₂ affects the phase while ₁ stems from the Taylor expansion of the energy (equation (18)) and the discrepancy mainly originates from the dephasing in time, as shown in figure 4. The amplitude of the oscillation in v_⊥, v_z and ɛ, on the other hand, can be predicted quite well by the analytical expression even for nonnegligible ₁, ₂, which provides us a way to crudely estimate the relative error. For example, the average and oscillatory parts of v_⊥ may be roughly estimated, respectively, as ∼mξ₁/ɛ and ∼mξ₂/ɛ, and therefore the relative error is approximately Δ_⊥ ∼ ξ₂/ξ₁. The relative error for v_z is $\sim {m}^{2}{\xi }_{1}{\xi }_{2}/\left({\bar{v}}_{z}{\varepsilon }^{2}\right)$, which is smaller than Δ_⊥. For the energy, according to (19), the oscillations are closely related to ₁, i.e. δɛ/ɛ ≈ ₁/2. If we plug in the simulation parameters, these estimations qualitatively explain that for the same small parameter ₁ and ₂, v_⊥ has the largest deviation. For figure 5(c), we can see that the analytical results predict a very good approximation when ₂ = 0 no matter how large ₁ is. The reason is that when p_x is zero the energy is constant (see equation (19)). Consequently, the phase plays no role and the approximation is quite good for the entire range of ₁ values presented in the figure.

In the non-periodic case, when the ratio ω₂/ω₁ is not an integer, it is convenient to define s_L ≡ s₁ + s₃, s_R ≡ s₂ − s₃. Hence, one may write

$$\begin{equation}{\mathcal{T}}_{y}=2\pi \sum\limits _{{s}_{\mathrm{L}}}\sum\limits _{{s}_{\mathrm{R}}}{\mathcal{M}}_{y}\delta \left({{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}\right),\end{equation} \tag{ 66 }$$

with ${{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}\equiv {\psi }_{np}-\frac{\varepsilon }{m}\left({s}_{\mathrm{L}}{\omega }_{1}+{s}_{\mathrm{R}}{\omega }_{2}\right)$. The matrix element takes the form

$$\begin{equation}{\mathcal{M}}_{y}=\sum\limits _{{s}_{3}}{B}_{0}\left(\mathbf{3}\right)\left[{\xi }_{1}{B}_{0}\left(\mathbf{2}\right){B}_{2}\left(\mathbf{1}\right)+{\xi }_{2}{B}_{0}\left(\mathbf{1}\right){B}_{2}\left(\mathbf{2}\right)\right].\end{equation} \tag{ 67 }$$

An analogous procedure may be applied for the other components as well, yielding

$$\begin{equation}{\mathcal{T}}_{\mu }=2\pi \sum\limits _{{s}_{\mathrm{L}}}\sum\limits _{{s}_{\mathrm{R}}}{\mathcal{M}}_{\mu }\left({s}_{\mathrm{L}},{s}_{\mathrm{R}},{\omega }^{\prime },\mathrm{cos}\enspace \theta \right)\delta \left({{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}\right),\end{equation} \tag{ 68 }$$

where

$$\begin{equation}{\mathcal{M}}_{t}=\sum\limits _{{s}_{3}}{B}_{0}\left(\mathbf{3}\right)\left[\frac{\varepsilon }{m}{B}_{0}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right)+\frac{{p}_{x}\omega {\xi }_{1}}{{\omega }_{1}\varepsilon }{B}_{1}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right)+\frac{{p}_{x}\omega {\xi }_{2}}{{\omega }_{2}\varepsilon }{B}_{0}\left(\mathbf{1}\right){B}_{1}\left(\mathbf{2}\right)\right],\end{equation} \tag{ 69 }$$

$$\begin{equation}{\mathcal{M}}_{x}=\sum\limits _{{s}_{3}}\left(\frac{{p}_{x}}{m}{B}_{0}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right){B}_{0}\left(\mathbf{3}\right)+{B}_{0}\left(\mathbf{3}\right)\left[{\xi }_{1}{B}_{0}\left(\mathbf{2}\right){B}_{1}\left(\mathbf{1}\right)+{\xi }_{2}{B}_{0}\left(\mathbf{1}\right){B}_{1}\left(\mathbf{2}\right)\right]\right),\end{equation} \tag{ 70 }$$

$$\begin{equation}{\mathcal{M}}_{z}=\sum\limits _{{s}_{3}}\left({B}_{0}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right)\left[\frac{{\bar{P}}_{z}}{m}{B}_{0}\left(\mathbf{3}\right)-\frac{m{\xi }_{1}{\xi }_{2}}{{\bar{v}}_{z}\varepsilon }{B}_{1}\left(\mathbf{3}\right)\right]+{B}_{0}\left(\mathbf{3}\right)\frac{{p}_{x}\omega }{\varepsilon }\left[\frac{{\xi }_{1}}{{\omega }_{1}}{B}_{1}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right)-\frac{{\xi }_{2}}{{\omega }_{2}}{B}_{0}\left(\mathbf{1}\right){B}_{1}\left(\mathbf{2}\right)\right]\right).\end{equation} \tag{ 71 }$$

As the squaring $\mathcal{T}$ does not mix terms associated with different s_L, s_R indices, the interference takes place only between terms included within $\mathcal{M}\left({s}_{\mathrm{L}},{s}_{\mathrm{R}}\right)$. Finally, the emitted intensity may be obtained by integrating (51) over the polar angle.

$$\begin{equation}\frac{\mathrm{d}I}{\mathrm{d}{\omega }^{\prime }\enspace \mathrm{d}\varphi }=\frac{\alpha m}{2\pi {\varepsilon }^{\prime }}\int \mathrm{d}\left(\mathrm{cos}\enspace \theta \right){\omega }^{\prime 2}\sum\limits _{{s}_{\mathrm{L}}}\sum\limits _{{s}_{\mathrm{R}}}{\left\vert \mathcal{M}\left({s}_{\mathrm{L}},{s}_{\mathrm{R}},{\omega }^{\prime },\mathrm{cos}\enspace \theta \right)\right\vert }^{2}\delta \left({{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}\right)\end{equation} \tag{ 72 }$$

where the identity ${\delta }^{2}\left({{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}\right)=\frac{{\tau }_{0}}{2\pi }\delta \left({{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}\right)$ is used. The proper interaction time is given by τ₀ = (m/ɛ)T₀. The condition imposed by the δ function, ${{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}=0$, determines the relation between cos θ and ω', φ

$$\begin{equation}1-\rho -{\bar{v}}_{z}\enspace \mathrm{cos}\enspace \theta ={\bar{v}}_{x}\enspace \mathrm{cos}\enspace \varphi \sqrt{1-{\mathrm{cos}}^{2}\enspace \theta }.\end{equation} \tag{ 73 }$$

Squaring and solving this equation one obtains two possible angles

$$\begin{equation}\mathrm{cos}\enspace {\theta }_{{\pm}}=\frac{{\bar{v}}_{z}\left(1-\rho \right){\pm}{\bar{v}}_{x}\enspace \mathrm{cos}\enspace \varphi \sqrt{{\Delta}}}{{\bar{v}}_{z}^{2}+{\bar{v}}_{x}^{2}\enspace {\mathrm{cos}}^{2}\enspace \varphi },\end{equation} \tag{ 74 }$$

where the following quantities were introduced

$$\begin{equation}{\Delta}\equiv {\bar{v}}_{z}^{2}+{\bar{v}}_{x}^{2}\enspace {\mathrm{cos}}^{2}\enspace \varphi -{\left(1-\rho \right)}^{2},\qquad \rho \equiv \frac{{\varepsilon }^{\prime }}{\varepsilon {\omega }^{\prime }}\left({s}_{\mathrm{L}}{\omega }_{1}+{s}_{\mathrm{R}}{\omega }_{2}\right).\end{equation} \tag{ 75 }$$

Notice that when squaring (73) a redundant solution may be added, which solves the equation

$$\begin{equation}1-\rho -{\bar{v}}_{z}\enspace \mathrm{cos}\enspace \theta =-{\bar{v}}_{x}\enspace \mathrm{cos}\enspace \varphi \sqrt{1-{\mathrm{cos}}^{2}\enspace \theta },\end{equation} \tag{ 76 }$$

rather than the original one. Thus, the solutions given in (74) are physical only when a positive results appear after substituting it into the right wing of (73). In quantitative terms, this condition reads

$$\begin{equation}\frac{1-\rho }{{\bar{v}}_{z}}{< }\mathrm{cos}\enspace \theta {\leqslant}1.\end{equation} \tag{ 77 }$$

A solution that does not meet this criterion is therefore excluded. Employing the δ function to perform the integration leads to

$$\begin{equation}\frac{\mathrm{d}I}{\mathrm{d}{\omega }^{\prime }\enspace \mathrm{d}\varphi }=\frac{\alpha \varepsilon {\omega }^{\prime 2}}{2\pi {\varepsilon }^{\prime }}\sum\limits _{i={\pm}}\sum\limits _{{s}_{\mathrm{L}}}\sum\limits _{{s}_{\mathrm{R}}}{\left\vert \mathcal{M}\left({s}_{\mathrm{L}},{s}_{\mathrm{R}},{\omega }^{\prime },{\theta }_{i}\right)\right\vert }^{2}{\left\vert \frac{\mathrm{d}{{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}}{\mathrm{d}\left(\mathrm{cos}\enspace \theta \right)}\right\vert }_{\theta ={\theta }_{i}}^{-1}.\end{equation} \tag{ 78 }$$

The reciprocal of the derivative of the δ function, required for the integration, reads

$$\begin{equation}{\left\vert \frac{\mathrm{d}{{\Omega}}_{{s}_{\mathrm{L}},{s}_{\mathrm{R}}}}{\mathrm{d}\left(\mathrm{cos}\enspace \theta \right)}\right\vert }^{-1}=\frac{m{\varepsilon }^{\prime }}{{\varepsilon }^{2}{\omega }^{\prime }}\kappa \quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\kappa \equiv \left\vert \frac{1}{{\bar{v}}_{x}\enspace \mathrm{cos}\enspace \varphi \enspace \mathrm{cot}\enspace \theta -{\bar{v}}_{z}}\right\vert .\end{equation} \tag{ 79 }$$

Plugging (79) into (78) the final result follows

$$\begin{equation}\frac{\mathrm{d}I}{\mathrm{d}{\omega }^{\prime }\enspace \mathrm{d}\varphi }=\frac{\alpha {\omega }^{\prime }{m}^{2}}{2\pi {\varepsilon }^{2}}\sum\limits _{i={\pm}}\sum\limits _{{s}_{\mathrm{L}}}\sum\limits _{{s}_{\mathrm{R}}}{\left\vert \mathcal{M}\left({s}_{\mathrm{L}},{s}_{\mathrm{R}},{\omega }^{\prime },{\theta }_{i}\right)\right\vert }^{2}\kappa \left({\theta }_{i}\right).\end{equation} \tag{ 80 }$$

For spinor particle the initial emission expression (51) is modified as follows

$$\begin{equation}\vert {\mathcal{T}}_{\mu }{\vert }^{2}\to \vert \mathcal{K}{\vert }^{2}\equiv -\left(\frac{{\varepsilon }^{\prime 2}+{\varepsilon }^{2}}{2\varepsilon {\varepsilon }^{\prime }}\right)\vert {\mathcal{T}}_{\mu }{\vert }^{2}+\frac{{\omega }^{\prime 2}}{2{\varepsilon }^{\prime 2}{\varepsilon }^{\prime 2}}\vert {\mathcal{T}}_{0}{\vert }^{2}.\end{equation} \tag{ 81 }$$

Therefore, the final results for scalars (80) is multiplied by $\left(\frac{{\varepsilon }^{\prime 2}+{\varepsilon }^{2}}{2\varepsilon {\varepsilon }^{\prime }}\right)$ and a second term is added

$$\begin{equation}\frac{\mathrm{d}I}{\mathrm{d}{\omega }^{\prime }\enspace \mathrm{d}\varphi }=\frac{\alpha {m}^{2}{\omega }^{\prime }}{4\pi {\varepsilon }^{5}{\varepsilon }^{\prime }}\sum\limits _{i={\pm}}\sum\limits _{{s}_{\mathrm{L}}}\sum\limits _{{s}_{\mathrm{R}}}\kappa \left({\theta }_{i}\right)\left[-{\varepsilon }^{2}\left({\varepsilon }^{2}+{\varepsilon }^{\prime 2}\right)\vert {\mathcal{M}}_{\mu }\left({s}_{\mathrm{L}},{s}_{\mathrm{R}}\right){\vert }^{2}+{\omega }^{\prime 2}{m}^{2}\vert {\mathcal{M}}_{0}\left({s}_{\mathrm{L}},{s}_{\mathrm{R}}\right){\vert }^{2}\right].\end{equation} \tag{ 82 }$$

Now, we consider the case of periodic motion when ω₂ = nω₁, with an integer n. As a result, the kinematic relation which follows from the δ function is modified. Using the relation between s_L, s_R and s₁, s₂, s₃ one obtains s_L ω₁ + s_R ω₂ → s_* ω₁, with the definition s_* ≡ s₁ + ns₂ − s₃(n − 1). Accordingly, ρ in (75) is replaced by $\rho =\frac{{\varepsilon }^{\prime }}{\varepsilon {\omega }^{\prime }}{s}_{{\ast}}{\omega }_{1}$. As a consequence of the periodicity, the summation over s₂ takes place inside the matrix element, similarly to s₃. Correspondingly, we have

$$\begin{equation}{\mathcal{M}}_{t}=\sum\limits _{{s}_{2}}\sum\limits _{{s}_{3}}{B}_{0}\left(\mathbf{3}\right)\left[\frac{\varepsilon }{m}{B}_{0}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right)+\frac{{p}_{x}\omega {\xi }_{1}}{{\omega }_{1}\varepsilon }{B}_{1}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right)+\frac{{p}_{x}\omega {\xi }_{2}}{{\omega }_{2}\varepsilon }{B}_{0}\left(\mathbf{1}\right){B}_{1}\left(\mathbf{2}\right)\right],\end{equation} \tag{ 83 }$$

$$\begin{equation}{\mathcal{M}}_{x}=\sum\limits _{{s}_{2}}\sum\limits _{{s}_{3}}\left(\frac{{p}_{x}}{m}{B}_{0}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right){B}_{0}\left(\mathbf{3}\right)+{B}_{0}\left(\mathbf{3}\right)\left[{\xi }_{1}{B}_{0}\left(\mathbf{2}\right){B}_{1}\left(\mathbf{1}\right)+{\xi }_{2}{B}_{0}\left(\mathbf{1}\right){B}_{1}\left(\mathbf{2}\right)\right]\right),\end{equation} \tag{ 84 }$$

$$\begin{equation}{\mathcal{M}}_{y}=\sum\limits _{{s}_{2}}\sum\limits _{{s}_{3}}{B}_{0}\left(\mathbf{3}\right)\left[{\xi }_{1}{B}_{0}\left(\mathbf{2}\right){B}_{2}\left(\mathbf{1}\right)+{\xi }_{2}{B}_{0}\left(\mathbf{1}\right){B}_{2}\left(\mathbf{2}\right)\right],\end{equation} \tag{ 85 }$$

$$\begin{equation}{\mathcal{M}}_{z}=\sum\limits _{{s}_{2}}\sum\limits _{{s}_{3}}\left({B}_{0}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right)\left[\frac{{\bar{P}}_{z}}{m}{B}_{0}\left(\mathbf{3}\right)-\frac{m{\xi }_{1}{\xi }_{2}}{{\bar{v}}_{z}\varepsilon }{B}_{1}\left(\mathbf{3}\right)\right]+{B}_{0}\left(\mathbf{3}\right)\frac{{p}_{x}\omega }{\varepsilon }\left[\frac{{\xi }_{1}}{{\omega }_{1}}{B}_{1}\left(\mathbf{1}\right){B}_{0}\left(\mathbf{2}\right)-\frac{{\xi }_{2}}{{\omega }_{2}}{B}_{0}\left(\mathbf{1}\right){B}_{1}\left(\mathbf{2}\right)\right]\right).\end{equation} \tag{ 86 }$$

Compared to the non-periodic case, the interference between different harmonics in the spectrum is much more complicated in the periodic case as there is a double summation inside the squaring of the matrix elements. The final result, analogous to (82) of the non-periodic case, is given by

$$\begin{equation}\frac{\mathrm{d}I}{\mathrm{d}{\omega }^{\prime }\enspace \mathrm{d}\varphi }=\frac{\alpha {m}^{2}{\omega }^{\prime }}{4\pi {\varepsilon }^{5}{\varepsilon }^{\prime }}\sum\limits _{i={\pm}}\sum\limits _{{s}_{{\ast}}}\kappa \left({\theta }_{i}\right)\left[-{\varepsilon }^{2}\left({\varepsilon }^{2}+{\varepsilon }^{\prime 2}\right)\vert {\mathcal{M}}_{{s}_{{\ast}}}^{\mu }{\vert }^{2}+{\omega }^{\prime 2}{m}^{2}\vert {\mathcal{M}}_{{s}_{{\ast}}}^{0}{\vert }^{2}\right].\end{equation} \tag{ 87 }$$

It is worth to point out that the periodic case is most likely to be observed in a short laser pulse, when the condition ω₂ = nω₁ can be fulfilled within the broad bandwidth of the laser pulse. We discuss this issue below.

In this subsection several quantities are explicitly evaluated for the particular case of vanishing initial transverse momentum, p_x = 0. It allows us to simplify the expressions and thus to obtain order of magnitude estimations which will prove useful later on. Substituting p_x = 0 to (74), the emitted photon angle reduces to

$$\begin{equation}\mathrm{cos}\enspace \theta =\frac{1-\rho }{{\bar{v}}_{z}}.\end{equation} \tag{ 88 }$$

Since ρ ≪ 1, the corresponding sine function is approximately given by $\mathrm{sin}\enspace \theta \approx \sqrt{1-\frac{1}{{\bar{v}}_{z}^{2}}+\frac{2\rho }{{\bar{v}}_{z}^{2}}}$. Substituting this expression to the Bessel arguments definitions (60) one obtains

$$\begin{equation}{z}_{1}=\frac{{\xi }_{1}{m}_{{\ast}}m}{{\omega }_{1}\varepsilon }\sqrt{u\left({u}_{s}-u\right)},\qquad {z}_{2}=\frac{{\xi }_{2}{m}_{{\ast}}m}{{\omega }_{2}\varepsilon }\sqrt{u\left({u}_{s}-u\right)}\end{equation} \tag{ 89 }$$

where we have defined ${u}_{s}\equiv \frac{2\varepsilon \left({s}_{\mathrm{L}}{\omega }_{1}+{s}_{\mathrm{R}}{\omega }_{2}\right)}{{m}_{{\ast}}^{2}}$ and the relation ${\bar{v}}_{z}^{2}=1-\frac{{m}_{{\ast}}^{2}}{{\varepsilon }^{2}}$ was employed. The maximal value of z₁, z₂ corresponds to u = u_s /2, namely

$$\begin{equation}{z}_{1}^{\mathrm{max}}=\frac{{\xi }_{1}\left({s}_{\mathrm{L}}{\omega }_{1}+{s}_{\mathrm{R}}{\omega }_{2}\right)}{{\omega }_{1}{m}_{{\ast}}}=\frac{{m}_{{\ast}}u{\xi }_{1}}{{\omega }_{1}\varepsilon },\qquad {z}_{2}^{\mathrm{max}}=\frac{{\xi }_{2}\left({s}_{\mathrm{L}}{\omega }_{1}+{s}_{\mathrm{R}}{\omega }_{2}\right)}{{\omega }_{2}{m}_{{\ast}}}\frac{{m}_{{\ast}}u{\xi }_{2}}{{\omega }_{2}\varepsilon }.\end{equation} \tag{ 90 }$$

In what follows we consider in detail the case where the copropagating beam is of relativistic intensity. It should be stressed that the spectrum may not be approximated by LCFA even though ξ₁ ≫ 1. The physical conditions and the nature of this specific LCFA violation is discussed in [41].

In the strong field regime the argument of the Bessel function in equation (58) can be the order of 10⁸ or even larger with the increasing of the laser field strength. This means the sum over the harmonics in the emission spectrum covers an extremely large region. In order to make the calculation feasible, we have employed an optimised scheme for the calculation, based on the logic proposed by Ritus and Sov [5].

It is well known that an ultrarelativistic particle emits mainly within a cone of angle ∼1/γ along its propagation direction. Hence, the emission angle θ may be approximated by the angle of the particle's momentum between P with respect to the z axis. Examining the classical momentum P, one observes that this angle lies in the range sin θ_d < sin θ < sin θ_u and its time-averaged value is sin θ_c , where

$$\begin{equation}\mathrm{sin}\enspace {\theta }_{c}\equiv \frac{\sqrt{{m}_{{\ast}}^{2}+{p}_{x}^{2}}}{\varepsilon },\qquad \mathrm{sin}\enspace {\theta }_{d}\equiv \frac{{p}_{x}\enspace \mathrm{cos}\enspace \varphi +m\left({\xi }_{1}-{\xi }_{2}\right)}{\varepsilon },\qquad \mathrm{sin}\enspace {\theta }_{u}\equiv \frac{{p}_{x}\enspace \mathrm{cos}\enspace \varphi +m\left({\xi }_{1}+{\xi }_{2}\right)}{\varepsilon }.\end{equation} \tag{ 91 }$$

In the case considered here, namely ξ ≫ 1, ξ₂ ≪ ξ₁, and due to p_x ≪ mξ₁ (see equation (20)), this range is very narrow and the angle may be crudely estimated according to the average value θ_c . Accordingly, one may show that the second term in the brackets appearing in the expression for ${z}_{1}^{x},{z}_{2}^{x}$ is negligible. As a result, the p_x = 0 expressions (60) provides an order of magnitude estimation for z₁, z₂, z₃. Plugging in sin θ_c ≈ m_*/ɛ, cos θ_c ≈ 1 one obtains

$$\begin{equation}{z}_{1}^{c}=\frac{{m}_{{\ast}}{\xi }_{1}u}{{\omega }_{1}\varepsilon },\qquad {z}_{2}^{c}=\frac{{m}_{{\ast}}{\xi }_{2}u}{{\omega }_{2}\varepsilon },\qquad {z}_{3}^{c}=\frac{m{\xi }_{1}{\xi }_{2}u}{{\bar{v}}_{z}{\omega }_{2}\varepsilon }.\end{equation} \tag{ 92 }$$

Notice that ${z}_{1}^{c},{z}_{2}^{c}$ coincide with the maximal value possible for these quantities, see equation (90). Furthermore, one may observe that since ${\bar{v}}_{z}\approx 1$ and m_* ≈ mξ₁ we have ${z}_{3}^{c}\approx {z}_{2}^{c}$.

In the following we take advantage of these relations in order to accelerate the harmonics summation (s_L, s_R, s₃) appearing in the final emission formula (82) as well as derive simplified validity conditions. We follow the logic presented by Ritus and Sov [5] for emission in a circularly polarized laser. Since u ∼ χ, these arguments may be much larger than 1. As a result, the number of harmonics contributing to the emission may be enormous, and an efficient way to carry out the summation is required. First, we replace the summation by integration. Second, since Bessel function of high order is maximal for z ≈ s and strongly suppressed for either z ≫ s or z ≪ s, the integration is centered around

$$\begin{equation}{s}_{\mathrm{L}}^{c}={z}_{1}^{c}+{z}_{3}^{c},\qquad {s}_{\mathrm{R}}^{c}={z}_{2}^{c}-{z}_{3}^{c},\qquad {s}_{3}^{c}={z}_{3}^{c}.\end{equation} \tag{ 93 }$$

In order to estimate the integration range, we define ${z}_{1}^{d},{z}_{2}^{d},{z}_{3}^{d}$ and ${z}_{1}^{u},{z}_{2}^{u},{z}_{3}^{u}$ analogously to ${z}_{1}^{c},{z}_{2}^{c},{z}_{3}^{c}$ appearing in equation (92) with θ = θ_d and θ = θ_u respectively. Accordingly, the upper and lower limits of the integration are respectively

$$\begin{equation}{s}_{\mathrm{L}}^{u}={z}_{1}^{u}+{z}_{3}^{u},\qquad {s}_{\mathrm{R}}^{u}={z}_{2}^{u}-{z}_{3}^{u},\qquad {s}_{3}^{u}={z}_{3}^{u}.\qquad {s}_{\mathrm{L}}^{d}={z}_{1}^{d}+{z}_{3}^{d},\qquad {s}_{\mathrm{R}}^{d}={z}_{2}^{d}-{z}_{3}^{d},\qquad {s}_{3}^{d}={z}_{3}^{d}.\end{equation} \tag{ 94 }$$

In mathematical terms, our improved summation scheme may be formulated as

$$\begin{equation}\sum\limits _{{s}_{\mathrm{L}}}\to {\int }_{{s}_{\mathrm{L}}^{d}}^{{s}_{\mathrm{L}}^{u}}d{s}_{\mathrm{L}},\underset{{s}_{\mathrm{R}}}{\sum }\to {\int }_{{s}_{\mathrm{R}}^{d}}^{{s}_{\mathrm{R}}^{u}}\mathrm{d}{s}_{\mathrm{R}},\qquad \sum\limits _{{s}_{3}}\to {\int }_{{s}_{3}^{d}}^{{s}_{3}^{u}}\mathrm{d}{s}_{3}.\end{equation} \tag{ 95 }$$

The replacement of the summation by integral in the calculation is appropriate only when s_u − s_d is large enough, which strongly depends on the chosen parameters.

In the following we present typical spectra in the strong field regime (ξ₁ = 20, ξ₂ = 0.3, p_x = 0), and use it to discuss the differences between the periodic and non-periodic cases derived above. Please note that the radiation reaction is neglected in the applied parameter regime, as the energy emitted during one laser cycle is very small compared with the electron energy.

Non-periodic case versus periodic case with p_x = 0. In figures 6(a) and (c), we consider the non-periodic case. The energy was chosen to be ɛ = 4m^*, so that the ratio ω₁/ω₂ is approximately 60.1. Since the radiation of an ultrarelativistic electron emitted to a certain direction originates from the vicinity of the location where the particle velocity points to the detector, it implies that the emission should depend on φ. However, due to the non-periodicity, we can see from figure 6(a) that the emission at a given φ takes place with different χ values in different ω₁ cycles. As a result of this χ-averaging, the difference between emissions at various φ disappears with long enough pulse of ξ₁, see the spectra in figure 6(c) for three different φ. The spectrum was evaluated with the aid of the non-periodic formula (82) together with (95).

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In the periodic case, figures 6(b) and (d), the electron energy is tuned ɛ = 4.02m^* to fulfil the integer ratio ω₂/ω₁ = 60. As opposed to the non-periodic case, here a particular value of χ parameter corresponds to the emission at a given φ at any period of the trajectory (see figure 6(b)), and therefore the emission depends on φ. In figure 6(d) the black, blue and red curves are calculated with different values of φ, respectively. One may see that these three curves significantly differ from each other.

Now we examine numerically the spectrum obtained for the non-periodic case, but with finite number of cycles in the laser pulse (as compared to the infinite pulse assumed by the analytical derivation). For the numerical calculations, we have evaluated equation (51) numerically, employing the numerical trajectory for the electron, as for realistic laser pulses the trajectory is not available analytically. In figure 7 full (hollow) circles designate ten cycles with φ corresponding to, respectively, the minimum (maximum) χ and full (hollow) squares are for five cycles with the same φ. First of all, the non-periodic spectrum, which represents averaging over φ, lies indeed in the middle between those curves, as expected. Secondly, one may see that the spectra for the finite laser pulse are far from the infinite pulse calculation. Moreover, the shorter the pulse is, the closer the results are to the periodic case. The reason is that the averaging out of the azimuthal dependence, as explained above, requires many cycles of interaction. The criterion which determines when one may employ the periodic formula is that the χ-averaging is not significant, namely

$$\begin{equation}N\left(n-{n}_{{\ast}}\right)\ll 2\pi \end{equation} \tag{ 96 }$$

where N is the number of cycles in the laser pulse, n = ω₂/ω₁ and n_* is the closest integer number to n. For the parameters considered above this quantity reads 0.5 and 1, respectively. Consequently, the periodic expression provides a good estimation to the final result for short laser pulse, provided that the condition (96) is fulfilled.

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It is worth to point out that there is a certain regime where the emission for χ_min is larger than for χ_max in figure 6(d). This is because in the region of χ_min along the electron's trajectory, the emission of the electron is not uniquely determined by the quantum parameter χ as commonly believed. This is because of the violation of the LCFA. In the formation length around χ_min, χ(t) changes rapidly and increases up to the order of χ_max and thus the emission is also similar near χ_max or even larger, see more discussions in reference [41].

Non-periodic case versus periodic case with p_x ≠ 0. Previously, we have discussed the emission of an electron in counterpropagating waves with vanishing transverse momentum. However, in a realistic experimental setup, the electrons in a beam always have non vanishing transverse momentum because of the angle spreading of the beam. In order to study the influence of the transverse momentum on the radiation process, we have in this section calculated the emission spectrum of an electron with p_x ≠ 0 for both non-periodic and periodic cases.

In figure 8, the spectra for p_x being 0.25% of the total energy have been investigated. Both of the spectra are not symmetric with respect to the azimuthal angle φ as the x-direction is favorable. For the non-periodic case, even the gradual shift of χ regarding the azimuthal angle still happens for p_x ≠ 0, the spectrum is nevertheless φ dependent because the transverse momentum breaks the symmetry. Furthermore, the spectrum for the periodic case with nonzero p_x has fringes with respect to φ. This means that the quantum parameter χ still has the similar dependence on the azimuthal angle like in figure 6(b).

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In the following we derive the validity conditions for the emission formula obtained in the previous section. For this purpose, we recall that the next order correction to the trajectory employed in this paper reads

$$\begin{equation}{\phi }_{1}\to {\phi }_{1}+{C}_{1}\enspace \mathrm{sin}\enspace {\phi }_{2}-{C}_{12}\enspace \mathrm{sin}\left({\phi }_{1}-{\phi }_{2}\right)\qquad {\phi }_{2}\to {\phi }_{2}+{C}_{2}\enspace \mathrm{sin}\enspace {\phi }_{1}+{C}_{12}\enspace \mathrm{sin}\left({\phi }_{1}-{\phi }_{2}\right).\end{equation} \tag{ 97 }$$

Let us substitute these modifications into the expression (55) for the phase ψ and examine the additional terms. Next, we take advantage of the identity

$$\begin{equation}\mathrm{sin}\left(\alpha \tau +{\beta }_{1}\enspace \mathrm{sin}\enspace {\kappa }_{1}\tau +{\beta }_{2}\enspace \mathrm{sin}\enspace {\kappa }_{2}\tau -\varphi \right)=\sum\limits _{{s}_{1}}\sum\limits _{{s}_{2}}\hspace{2pt}{J}_{{s}_{1}}\left({\beta }_{1}\right){J}_{{s}_{2}}\left({\beta }_{2}\right)\mathrm{sin}\left[\left(\alpha +{s}_{1}{\kappa }_{1}+{s}_{2}{\kappa }_{2}\right)\tau -\varphi \right].\end{equation} \tag{ 98 }$$

Please note that κ here is just a parameter and not related to equation (79). Consequently, sin, cos functions in (55) are replaced according to

$$\begin{equation}\mathrm{cos}\enspace \alpha \tau \to \qquad {\mathcal{I}}_{1}\left({\Lambda}\right)\equiv \alpha \int \mathrm{d}\tau \enspace \mathrm{sin}\left(\alpha \tau +{\beta }_{1}\enspace \mathrm{sin}\enspace {\kappa }_{1}\tau +{\beta }_{2}\enspace \mathrm{sin}\enspace {\kappa }_{2}\tau -\varphi \right),\end{equation} \tag{ 99 }$$

$$\begin{equation}\mathrm{sin}\enspace \alpha \tau \to \qquad {\mathcal{I}}_{2}\left({\Lambda}\right)\equiv \alpha \int \mathrm{d}\tau \enspace \mathrm{cos}\left(\alpha \tau +{\beta }_{1}\enspace \mathrm{sin}\enspace {\kappa }_{1}\tau +{\beta }_{2}\enspace \mathrm{sin}\enspace {\kappa }_{2}\tau -\varphi \right)\end{equation} \tag{ 100 }$$

with Λ denoting $\left(\alpha ,{\beta }_{1},{\kappa }_{1},{\beta }_{2},{\kappa }_{2},\varphi \right)$. Using (98) one obtains

$$\begin{equation}{\mathcal{I}}_{1}=-\sum\limits _{{s}_{1}}\sum\limits _{{s}_{2}}\frac{{J}_{{s}_{1}}\left({\beta }_{1}\right){J}_{{s}_{2}}\left({\beta }_{2}\right)}{1+\left({s}_{1}{\kappa }_{1}+{s}_{2}{\kappa }_{2}\right)/\alpha }\mathrm{cos}\left[\left(\alpha +{s}_{1}{\kappa }_{1}+{s}_{2}{\kappa }_{2}\right)\tau -\varphi \right],\end{equation} \tag{ 101 }$$

$$\begin{equation}{\mathcal{I}}_{2}=\sum\limits _{{s}_{1}}\sum\limits _{{s}_{2}}\frac{{J}_{{s}_{1}}\left({\beta }_{1}\right){J}_{{s}_{2}}\left({\beta }_{2}\right)}{1+\left({s}_{1}{\kappa }_{1}+{s}_{2}{\kappa }_{2}\right)/\alpha }\mathrm{sin}\left[\left(\alpha +{s}_{1}{\kappa }_{1}+{s}_{2}{\kappa }_{2}\right)\tau -\varphi \right].\end{equation} \tag{ 102 }$$

In the previous section the trigonometric identity ${z}_{1}^{x}\enspace \mathrm{cos}\enspace {\phi }_{1}+{z}_{1}^{y}\enspace \mathrm{sin}\enspace {\phi }_{1}={z}_{1}\enspace \mathrm{sin}\left({\phi }_{1}-{\varphi }_{1}\right)$ was employed, where z₁, φ₁ are given by (58) respectively. Analogously, in this case we have

$$\begin{equation}{z}_{1}^{x}{\mathcal{I}}_{2}\left({{\Lambda}}_{0}\right)+{z}_{1}^{y}{\mathcal{I}}_{1}\left({{\Lambda}}_{0}\right)={z}_{1}{\mathcal{I}}_{2}\left({{\Lambda}}_{1}\right)\end{equation} \tag{ 103 }$$

where

$$\begin{equation}{{\Lambda}}_{0}=\left(\frac{{\omega }_{1}\varepsilon }{m},{C}_{1},\frac{{\omega }_{2}\varepsilon }{m},-{C}_{12},\frac{\left({\omega }_{1}-{\omega }_{2}\right)\varepsilon }{m},0\right),\qquad {{\Lambda}}_{1}=\left(\frac{{\omega }_{1}\varepsilon }{m},{C}_{1},\frac{{\omega }_{2}\varepsilon }{m},-{C}_{12},\frac{\left({\omega }_{1}-{\omega }_{2}\right)\varepsilon }{m},{\varphi }_{1}\right).\end{equation} \tag{ 104 }$$

As a result, the modified phase may be written as

$$\begin{equation}\psi ={\psi }_{np}\tau -{z}_{1}{\mathcal{I}}_{2}\left({{\Lambda}}_{1}\right)-{z}_{2}{\mathcal{I}}_{2}\left({{\Lambda}}_{2}\right)-{z}_{3}{\mathcal{I}}_{2}\left({{\Lambda}}_{3}\right)\end{equation} \tag{ 105 }$$

where

$$\begin{equation}{{\Lambda}}_{2}=\left(\frac{{\omega }_{2}\varepsilon }{m},0,0,{C}_{12},\frac{\left({\omega }_{1}-{\omega }_{2}\right)\varepsilon }{m},{\varphi }_{2}\right),\qquad {{\Lambda}}_{3}=\left(\frac{\left({\omega }_{1}-{\omega }_{2}\right)\varepsilon }{m},{C}_{1},\frac{{\omega }_{2}\varepsilon }{m},-2{C}_{12},\frac{\left({\omega }_{1}-{\omega }_{2}\right)\varepsilon }{m},0\right).\end{equation} \tag{ 106 }$$

Let us estimate the neglected contribution to the phase, namely the difference between (59) and (105). For the sake of simplicity, we split the corrections to three contributions, Δψ₁, Δψ₂, Δψ₃ associated with z₁, z₂, z₃, respectively.

$$\begin{equation}{\Delta}\psi ={\Delta}{\psi }_{1}+{\Delta}{\psi }_{2}+{\Delta}{\psi }_{3}.\end{equation} \tag{ 107 }$$

In explicit terms, the corrections take the form ${\Delta}{\psi }_{1}=-{z}_{1}\left[{\mathcal{I}}_{2}\left({{\Lambda}}_{1}\right)-\mathrm{sin}\left({\phi }_{1}-{\varphi }_{1}\right)\right]$, and for Δψ₂, Δψ₃ we have z₁ → z₂, z₃. Since C₁, C₁₂ ≪ 1, we consider only first order corrections, namely s₁ = 0, s₂ = ±1 and s₁ = ±1, s₂ = 0. Therefore, one readily obtains

$$\begin{align}{\mathcal{I}}_{2}\left({{\Lambda}}_{1}\right)-\mathrm{sin}\left(\alpha \tau -\varphi \right)& \approx \frac{{\beta }_{1}}{2}\left(\frac{1}{1+{\kappa }_{1}/\alpha }\mathrm{sin}\left[\left(\alpha +{\kappa }_{1}\right)\tau -\varphi \right]-\frac{1}{1-{\kappa }_{1}/\alpha }\mathrm{sin}\left[\left(\alpha -{\kappa }_{1}\right)\tau -\varphi \right]\right)\\ & \quad +\left({\beta }_{1},{\kappa }_{1}\to {\beta }_{2},{\kappa }_{2}\right)\end{align} \tag{ 108 }$$

where J₁(β) = −J₋₁(β) ≈ β/2 was used. Using equations (104) and (108) yields

$$\begin{equation}{\Delta}{\psi }_{1}=-{\eta }_{1}\left[\mathrm{sin}\left({\phi }_{1}+{\phi }_{2}-{\varphi }_{1}\right)+\mathrm{sin}\left({\phi }_{1}-{\phi }_{2}-{\varphi }_{1}\right)\right]+{\eta }_{2}\left[\mathrm{sin}\left({\phi }_{2}-{\varphi }_{1}\right)-\mathrm{sin}\left(2{\phi }_{1}-{\phi }_{2}-{\varphi }_{1}\right)\right]\end{equation} \tag{ 109 }$$

where ω₁/ω₂ ≪ 1 was employed. Analogously, for the other contribution one finds

$$\begin{align}{\Delta}{\psi }_{2}& =-{\eta }_{3}\enspace \mathrm{sin}\left({\phi }_{1}-{\varphi }_{2}\right)-{\eta }_{4}\enspace \mathrm{sin}\left(2{\phi }_{2}-{\phi }_{1}-{\varphi }_{2}\right),\\ {\Delta}{\psi }_{3}& =-{\eta }_{5}\enspace \mathrm{sin}\left({\phi }_{1}-2{\phi }_{2}\right)+{\eta }_{6}\enspace \mathrm{sin}\enspace {\phi }_{1}+{\eta }_{7}\enspace \mathrm{sin}\left[2\left({\phi }_{1}-{\phi }_{2}\right)\right]+{\eta }_{8}\tau .\end{align} \tag{ 110 }$$

The following coefficients were defined

$$\begin{align}& {\eta }_{1}\equiv \frac{{z}_{1}{C}_{1}{\omega }_{1}}{2{\omega }_{2}},\qquad {\eta }_{2}\equiv \frac{{z}_{1}{C}_{12}{\omega }_{1}}{2{\omega }_{2}},\qquad {\eta }_{3}\equiv \frac{{z}_{2}{C}_{12}{\omega }_{2}}{2{\omega }_{1}},\qquad {\eta }_{4}\equiv \frac{{z}_{2}{C}_{12}}{4},\qquad {\eta }_{5}\equiv \frac{{z}_{3}{C}_{1}}{4},\\ & {\eta }_{6}\equiv \frac{{z}_{3}{C}_{1}{\omega }_{2}}{2{\omega }_{1}},\qquad {\eta }_{7}\equiv \frac{{z}_{3}{C}_{12}}{2},{\eta }_{8}\equiv \frac{\left({\omega }_{1}-{\omega }_{2}\right)\varepsilon {z}_{3}{C}_{12}}{m}.\end{align}$$

In order to formulate the general validity condition, we notice that the phase (59) contains a linear term with low (ω₁) and high (ω₂, ω₂ − ω₁) frequencies. Therefore, we require that the coefficients of the high frequency corrections, will be smaller as compared to z₂, z₃. Similarly, the coefficients of the low frequency should be lower than z₁, and the one corresponding to the linear term smaller than ψ_np . Hence the general validity condition may be cast in the form

$$\begin{equation}{\eta }_{1},{\eta }_{2},{\eta }_{4},{\eta }_{5},{\eta }_{7}\ll {z}_{2},{z}_{3}\qquad {\eta }_{3},{\eta }_{6}\ll {z}_{1}\qquad {\eta }_{8}\ll {\psi }_{np}.\end{equation} \tag{ 111 }$$

We call attention to the fact that these conditions depend on the emitted photon properties ω, θ, φ. As a result, for given interaction parameters (laser amplitudes, particle energy), part of the spectrum may be described by our analytical expression whereas a different part may exhibit deviations. Hence, one should verify that (111) holds for the entire spectral range of interest. In the strong field case, however, the situation is much simplified and simple criteria are derived, which hold for the entire spectrum.

Let us consider explicitly the strong field regime (ξ₁ ≫ 1). As explained in section 3.4, in this regime the emission is restricted to a limited angle range, for which the Bessel coefficients may be approximated by ${z}_{1}^{c},{z}_{2}^{c},{z}_{3}^{c}$. Substituting these expressions to the requirement (111) and employing the trajectory validity conditions in section 2.4 as well as the approximation $\frac{{\omega }_{2}}{{\omega }_{1}}=\frac{1+{\bar{v}}_{z}}{1-{\bar{v}}_{z}}\approx \frac{4{\varepsilon }^{2}}{{m}_{{\ast}}^{2}}$, we find that η₄, η₅, η₇, η₈ obey (111) by definition. Employing equation (92) as well as the expressions for C₁, C₁₂, the validity condition is simplified to

$$\begin{equation}\frac{{\eta }_{1}}{{z}_{2}}=\frac{{p}_{x}{\xi }_{1}}{2{m}_{{\ast}}{\xi }_{2}}\ll 1,\qquad \frac{{\eta }_{6}}{{z}_{1}}=\frac{{p}_{x}{\xi }_{2}}{2{m}_{{\ast}}{\xi }_{1}}\ll 1\enspace ,\qquad \frac{{\eta }_{2}}{{z}_{2}}=\frac{{m}^{2}{\xi }_{1}^{2}}{2\varepsilon }\ll 1,\qquad \frac{{\eta }_{3}}{{z}_{1}}=\frac{{m}^{2}{\xi }_{2}^{2}}{2\varepsilon }\ll 1.\end{equation} \tag{ 112 }$$

The last three conditions are automatically fulfilled according to the validity conditions for the trajectory. Hence, only a single additional condition, corresponding to η₁ in equation (112), is required to validate the applied formalism:

$$\begin{equation}\frac{{p}_{x}}{2m{\xi }_{2}}\ll 1.\end{equation} \tag{ 113 }$$

As demonstrated above, the analytical approximation depends on several criteria being fulfilled. In the following, we examine in detail the strong field case, where the number of quantities required to be low is relatively small, allowing for a tractable study of the error. The main quantities, which stem from the trajectory approximation, are ₁, ₂ given in section 2.4. In the following we investigate systematically and quantitatively the relation between these parameters and the corresponding error. For the sake of this purpose, a new quantity is introduced

$$\begin{equation}{\Delta}=\frac{{{\Delta}}_{1}+{{\Delta}}_{2}+{{\Delta}}_{3}}{3},\qquad {{\Delta}}_{i}\equiv 100\frac{\vert {I}_{n}\left({u}_{i}\right)-{I}_{a}\left({u}_{i}\right)\vert }{{I}_{n}\left({u}_{i}\right)}\end{equation} \tag{ 114 }$$

measuring the relative difference Δ (in percent) between the analytical (I_a) and numerical (I_n) results. It is evaluated in three points u_i on the spectrum. Figure 9 shows the deviation Δ as a function of ₁, assuming that ₂ vanishes. One can see that error grows monotonically, and that ₁ = 0.1 yields a deviation of 15%. In order to examine the influence of ₂, the calculation was generalized to two dimensions, and the results are presented in table 2. One may see that the impact of ₂ is significantly smaller as compared to ₁. The results presented in this subsection provides quantitative information which may be valuable when applying our expressions in practice.

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Table 2. The relative difference between the analytical and numerical results as a function of the parameters ₁ and ₂.

	2
1	0.03	0.06	0.09	0.12	0.15
0.03	3.01	10.25	13.85	18.35	25.46
0.06	3.05	10.60	14.19	19.10	26.19
0.09	3.10	11.00	14.72	19.83	26.99
0.12	3.19	11.40	15.57	20.56	27.81
0.15	3.32	11.80	16.79	21.45	28.90

The analytical derivation presented above assumes that the laser fields are monochromatic plain waves. This approximation is appropriate for long pulses (dozens of cycles) which are focused on large spots (radius of dozens of wavelengths). However, realistic pulses tend to be short and tightly focused, in order to maximize the obtained intensity for a given pulse energy. Therefore, for practical reasons it is highly important to thoroughly examine the dependence of the emission on pulse duration and focal size. In particular, we wish to establish qualitatively which spectral features are affected by shortening/focusing the laser pulse, what is the amplitude of the deviation and to find the conditions for which the spectrum recovers the analytical result.

In order to specify the spatial and temporal shape for the realistic laser pulse, the following quantities are introduced

$$\begin{equation}X\equiv \frac{x}{{w}_{0}},\qquad Y\equiv \frac{y}{{w}_{0}},\qquad Z\equiv \frac{z}{{z}_{r}},\qquad {z}_{r}\equiv \frac{\omega {w}_{0}^{2}}{2},\qquad f\equiv \frac{\mathrm{i}}{\mathrm{i}+Z},\qquad \rho \equiv \sqrt{{X}^{2}+{Y}^{2}}.\end{equation} \tag{ 115 }$$

The vector potential corresponding to this pulse reads [64]

$$\begin{equation}{A}_{x}={A}_{0}g\left(k\cdot x\right)f{\mathrm{e}}^{-f{\rho }^{2}}\enspace {\text{e}}^{\text{i}k\cdot x}\left(1+{{\epsilon}}^{2}\left[\frac{f}{2}-\frac{{f}^{3}{\rho }^{4}}{4}\right]+{{\epsilon}}^{4}\left[\frac{3{f}^{2}}{8}-\frac{3{f}^{4}{\rho }^{4}}{16}-\frac{{f}^{5}{\rho }^{6}}{8}+\frac{{f}^{6}{\rho }^{8}}{32}\right]\right)\end{equation} \tag{ 116 }$$

where A₀ denotes the amplitude. Since we consider a circular polarization, the y component is given by A_y = iA_x . The EM fields can thus be derived from the above vector potential by

$$\begin{equation}\boldsymbol{E}=-\mathrm{i}\omega \boldsymbol{A}-\frac{\mathrm{i}}{\omega }\mathbf{\nabla }\left(\mathbf{\nabla }\cdot \boldsymbol{A}\right),\qquad \boldsymbol{B}=\mathbf{\nabla }{\times}\boldsymbol{A}.\end{equation} \tag{ 117 }$$

In the calculation below, we choose a sin²-function with σ₀ denoting the pulse length for g(k ⋅ x) as the temporal envelope. Figure 10 depicts the angle integrated emission of a particle interacting with pulses with normalized amplitudes ξ₁ = 12.5, ξ₂ = 0.1, ɛ = 80m, respectively. The solid line stands for the analytical expression. Numerical calculations corresponding to variety of pulse durations and focal radii were carried out as well. For the sake of comparison, we wanted to keep the energy of the particle in the main part of the pulse identical for all compared cases. For this purpose, the initial electron energy was zero and its initial location z₀ was tuned, namely the distance to the beginning of the ξ₁ and ξ₂ beams. From an experimental point of view, it may be realized by placing atoms which are ionized by the laser field.

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As expected, the analytical formula coincides fairly well with the numerical calculation for a long pulse with large focus w₀ = 50λ, σ₀ = 20T, with λ and T as the laser wave length and laser period, respectively. Let us examine the influence of the temporal width first. Decreasing the duration to σ = 10T does not change much the spectrum. However, for ultrashort pulses (full circles) with σ = 5T, the emission significantly increases. This may be explained by the fact that the rapid rise of the pulse is accompanied by stronger acceleration and enhanced χ value. Moreover, the shorter the pulse is, the larger the edge effect will be in the emission spectrum. This edge effect will induce deviations of the spectrum from the LCFA predictions and enhance the emission, especially in the high energy domain [41].

As for the spatial focusing, one observes an opposite trend. Namely, a small spot results in a significant decrease in the emitted spectrum, as well as in a deformation of its spectral shape. We suggest that this outcome stems from the fact that tightly focused beams rapidly expel the particle from the focus due to the transverse ponderomotive force. Furthermore, one may notice that even moderate focusing, w₀ = 20λ, results in a considerable deviation from the one dimensional case. Thus, figure 10 shows that finite duration yields significant deviation from the analytical expression only for ultrashort pulses, whereas the focal radius has greater influence and should be fairly large in order to recover the theoretical result.