Equations of electron interaction in the pulsed field of two laser waves

Consider the interaction of two non-relativistic electrons moving toward each other along the x axis in the field of two linearly polarized pulsed electromagnetic waves. Waves propagate perpendicularly to each other. The first wave propagates along the z axis, the second wave propagates along the x axis (see Fig. 1).

Fig. 1.

Interaction kinematics of two classical electrons in the field of two light mutually perpendicular waves.

The strengths of the electric and magnetic fields are given in the following form:

(1)$${\bf E}(t,z_j,x_j) = {\bf E}_1(t,z_j) + {\bf E}_2(t,x_j),$$

(2)$$\eqalign{ {\bf E}_1(t,z_j) = E_{01} \cdot \exp \left[ { - {\left( {\displaystyle{{{\rm \varphi} _{1j} - {\rm \delta} {\rm \tau} _1} \over {{\rm \omega} _1t_1^{}}}} \right)}^2} \right]\cos {\rm \varphi} _{1j} \cdot {\bf e}_x, & \cr \quad {\rm \varphi} _{1j} = ({\rm \omega} _1t - k_1z_j),&} $$

(3)$$\eqalign{ {\bf E}_2(t,x_j) = E_{02} \cdot \exp \left[ { - {\left( {\displaystyle{{{\rm \varphi} _{2j} - {\rm \delta} {\rm \tau} _2} \over {{\rm \omega} _2t_2}}} \right)}^2} \right]\cos {\rm \varphi} _{2j} \cdot {\bf e}_y, & \cr \quad {\rm \varphi} _{2j} = ({\rm \omega} _2t - k_2x_j), & } $$

(4)$${\bf H}(t,z_j,x_j) = {\bf H}_{\bf 1}(t,z_j) + {\bf H}_2(t,x_j),$$

(5)$${\bf H}_1(t,z_j) = H_{01} \cdot \exp \left[ { - {\left( {\displaystyle{{{\rm \varphi} _{1j} - {\rm \delta} {\rm \tau} _1} \over {{\rm \omega} _1t_1}}} \right)}^2} \right]\cos {\rm \varphi} _{1j} \cdot {\bf e}_y,$$

(6)$${\bf H}_2(t,x_j) = H_{02} \cdot \exp \left[ { - {\left( {\displaystyle{{{\rm \varphi} _{2j} - {\rm \delta} {\rm \tau} _2} \over {{\rm \omega} _2t_2}}} \right)}^2} \right]\cos {\rm \varphi} _{2j} \cdot {\bf e}_z,$$

where φ_ij are phases of the corresponding wave (i = 1, 2) and corresponding electron (j = 1, 2); E _0i and H _0i are the strength of the electric and magnetic fields in the pulse peak, respectively; δτ_i are phase shifts of pulse peaks of the first and second waves; t _i and ω_i are the pulse duration and frequency of the first and the second waves; ${\bf e}_x$, ${\bf e}_y$, ${\bf e}_z$ are unit vectors directed along the x, y, and z axes.

It is a well-known fact that in the frame of the dipole approximation (k = 0) and without taking into account the terms of the order v/c ≪ 1, the particle interaction with the plane-wave field does not affect the particle relative motion in the center-of-mass system. Thereby, we consider particle motion in the laser field beyond the dipole approximation and an accuracy of quantities of the order v/c ≪ 1 (v is the relative velocity). Newton equations for motion of two identically charged particles with the mass m and charge e (e = e ₁ = e ₂) in the pulsed field of two mutually perpendicular laser waves 1–6 are determined by the following expressions:

(7)$$m{\ddot{\bf r}}_1 = - \left \vert e \right \vert \left[ {{\bf E}\left( {t,z_1,x_1} \right) + \displaystyle{1 \over c}{\dot{\bf r}}_1 \times {\bf H}\left( {t,z_1,x_1} \right)} \right] - \displaystyle{{e^2} \over {{\left \vert {{\bf r}_2 - {\bf r}_1} \right \vert} ^3}}\left( {{\bf r}_2 - {\bf r}_1} \right),$$

(8)$$m{\ddot{\bf r}}_2 = - \left \vert e \right \vert \left[ {{\bf E}\left( {t,z_2,x_2} \right) + \displaystyle{1 \over c}{\dot{{\bf r}}}_2 \times {\bf H}\left( {t,z_2,x_2} \right)} \right] + \displaystyle{{e^2} \over {{\left \vert {{\bf r}_2 - {\bf r}_1} \right \vert} ^3}}\left( {{\bf r}_2 - {\bf r}_1} \right),$$

where ${\bf r}_1$ and ${\bf r}_2$ are electron radius-vectors.

Hereafter, the wave frequencies are the same:

$${\rm \omega}_1 = {\rm \omega}_2 = {\rm \omega}, \quad \vert{\bf k}_1\vert = \vert{\bf k}_2\vert = k = {\rm \omega}/c = {\lambdabar}^{-1}.$$

Subsequent consideration should be carried out in the center-of-mass system:

(9)$${\bf r} = {\bf r}_2 - {\bf r}_1,\quad {\bf R} = \displaystyle{1 \over 2}\left( {{\bf r}_2 + {\bf r}_1} \right).$$

The following equations are for the particle relative motion in the center-of-mass system:

(10)$$m{\ddot{\bf r}} = \displaystyle{{2e^2} \over {{\left \vert {\bf r} \right \vert} ^3}}{\bf r} - \left \vert e \right \vert \left( {{\rm M}_x{\bf e}_x + {\rm M}_y{\bf e}_y + {\rm M}_z{\bf e}_z} \right);$$

(11)$$\left\{ {\matrix{ {{\rm M}_x = f_1\left[ {2\sin ({\rm \omega} t)\sin \left( {k\displaystyle{{r_z} \over 2}} \right) - \displaystyle{1 \over c}{\dot r}_z\cos ({\rm \omega} t)} \right] + \displaystyle{1 \over c}f_2{\dot r}_y\cos ({\rm \omega} t)} \hfill \cr {{\rm M}_y = f_2\left[ {2\sin ({\rm \omega} t)\sin \left( {k\displaystyle{{r_x} \over 2}} \right) - \displaystyle{1 \over c}{\dot r}_x\cos ({\rm \omega} t)} \right]\quad {\rm}} \hfill \cr {{\rm M}_z = \displaystyle{1 \over c}f_1{\dot r}_x\cos ({\rm \omega} t){\rm}} \hfill \cr}} \right..$$

(12)$$f_1 = E_{01}\exp \left[ { - \displaystyle{{{\left( {{\rm \omega} t - {\rm \delta} {\rm \tau} _1} \right)}^2} \over {{\left( {{\rm \omega} t_1} \right)}^2}}} \right],\quad f_2 = E_{02}\exp \left[ { - \displaystyle{{{\left( {{\rm \omega} t - {\rm \delta} {\rm \tau} _2} \right)}^2} \over {{\left( {{\rm \omega} t_2} \right)}^2}}} \right].$$

Note that the equations for relative motion (10), (11) are written beyond the dipole approximation (k ≠ 0) and an accuracy of term of order v/c ≪ 1. Note, small influence of external strong-pulsed laser field on radius-vector of the center-of-mass motion was shown in Starodub et al. (Reference Starodub, Roshchupkin and Dubov2016). Thereby, it is necessary to study only the relative motion.

Equations (10), (11) can be written in the dimensionless form:

(13)$$b{\ddot{\bf {\xi}}} = {\bf F},\quad {\bf F} = {\rm \beta} \cdot \displaystyle{{\bf { \xi}} \over {\bf {\left \vert {\bf {\xi}} \right \vert} ^3}} - (N_x{\bf e}_x + N_y{\bf e}_y + N_z{\bf e}_z),$$

(14)$$\left\{ {\matrix{ {N_x = {\rm \eta} _1f_1^{} \left[ {\sin ({\rm \tau} )\sin \left( {\displaystyle{{{\rm \xi} _z} \over 2}} \right) - \displaystyle{{{\dot {\rm \xi}} _z} \over 2}\cos ({\rm \tau} )} \right] + {\rm \eta} _2f_2^{} {\dot {\rm \xi}} _y\cos ({\rm \tau} )} \hfill \cr {N_y = {\rm \eta} _2f_2^{} \left[ {\sin ({\rm \tau} )\sin \left( {\displaystyle{{{\rm \xi} _x} \over 2}} \right) - \displaystyle{{{\dot {\rm \xi}} _x} \over 2}\cos ({\rm \tau} )} \right]{\rm}} \hfill \cr {N_z = {\rm \eta} _1f_1^{} \displaystyle{{{\dot {\rm \xi}} _x} \over 2}\cos ({\rm \tau} ){\rm}} \hfill \cr}} \right.,$$

where,

(15)$${\bf \xi} = k{\bf r} = {\bf r} / {\lambdabar}\,, \;\;\;{\rm \tau} = {\rm \omega} t,\;\;{\rm \tau} _{1,2} = {\rm \omega} t_{1,2};$$

(16)$$\eqalign{& {\rm \eta} _i = \displaystyle{{\left \vert e \right \vert E_{0i}} \over {{\rm \mu} c{\rm \omega}}}, \;\quad {\rm \beta} = \displaystyle{{{e^2} / {\lambdabar}} \over {{\rm \mu} c^2}},\quad {\rm \mu} = {m / 2}, \cr & \quad f_i^{} = \exp \left( { - \displaystyle{{{\left( {{\rm \tau} - {\rm \delta} {\rm \tau} _i} \right)}^2} \over {{\rm \tau} _i^2}}} \right),\quad i = 1,\;2.} $$

Here, ${\bf { \xi}} $ is the radius-vector of the relative distance between electrons in unit of the wavelength, the parameters η_1,2 are numerically equal to the ratio of the oscillation velocity of an electron in the peak of a pulse of the first or second wave to the velocity of light c (hereinafter, should consider parameters η_1,2 as oscillation velocities); the parameter β is numerically equal to the ratio of the energy of Coulomb interaction of electrons with the reduced mass μ at the wavelength to the particle rest energy.

The pulse duration exceeds considerably the period of wave rapid oscillation (~ω⁻¹) for a majority of modern pulsed lasers:

(17)$${\rm \tau} _{1,2} \gg 1.$$

Consequently, the relative distance between electrons should be averaged over the period of wave rapid oscillation:

(18)$$\bar {\rm \xi} = \displaystyle{1 \over {2{\rm \pi}}} \int\limits_0^{2{\rm \pi}} {{\rm \xi} \cdot d{\rm \tau}}. $$

It is worth to note that expressions (13), (14) consider interaction with the Coulomb field and the pulsed-wave field strictly, and do not have the analytical solution. For subsequent analysis, all equations will be studied numerically.

Electrons initial relative coordinates and velocities are the following:

(19)$$\eqalign{& {\rm \xi} _{x0} = 2,\quad {\rm \xi} _{y0} = 0,\quad \;{\rm \xi} _{z0} = 0, \cr & {\dot {\rm \xi}} _{x0} = - 1.7 \cdot 10^{ - 3}{\rm,} \quad {\dot {\rm \xi}} _{y0} = 0,\quad {\dot {\rm \xi}} _{z0} = 0.} $$

The interaction time is ${\rm \tau} = 1200\;({\rm \tau} \in \left[ { - 600 \div 600} \right])$, (t = 600 fs) and it was increased, if necessary for more clear results. Frequencies of waves are ω₁ = ω₂ = 2 Ps⁻¹ (${\lambdabar} = 0.15{\rm \mu} {\rm m}$), pulse durations are τ₁ = τ₂ = 600 (t ₁ = t ₂ = 300 fs). Field intensities (oscillations velocities η₁, η₂) are varied. Phase shifts are vary within ${\rm \delta} {\rm \tau} _{1,2} \in \left[ { - 600 \div 600} \right]$ and step is h = 50. Initial conditions are the same as in (9). That allows to estimate the influence of phase shifts on relative motion of electrons and compare results. Note, in the work (9) the parameter of the phase shift of a pulse of a wave (δτ_1,2 = 0) was absent, and pulse peaks of both waves were in moment τ = 0. Initial coordinates and velocities of electrons are chosen, so that at the point τ = 0 electrons were in maximum approach (the Coulomb force was maximum). In this work, the pulse peaks of waves can be maximum at any moment of time (unlike the previous publication (9)) and it leads to significant change in the behavior of electron interaction. Numerical solving of equations for relative motion (13) results to several cases.

Anomalous repulsion of electrons

The case when the oscillation velocity of the first wave is greater than the initial velocity of electrons (${\rm \eta} _1 \gt \dot {\rm \xi} _0$), and oscillation velocity of the second wave considerably exceeds the initial velocity (${\rm \eta} _2 \gg \dot {\rm \xi} _0$).

Calculations over all values of phase shifts allowed to find out areas of anomalous repulsion of electrons. In these areas, electrons can scatter at very long distances exceeding the distance of electron scattering without an external field in hundreds of times (see Fig. 2a, 2b). Let us designate the final distance at which electrons scatter in the time moment as τ_Cfinal = 600: without an external field as $\bar {\rm \xi} _{{\rm Cfinal}} = 2$; in the external field, when δτ₁ = 0,   δτ₂ = 0 as $\bar {\rm \xi} ^{(0)}_{{\rm final}} $; in an external field, when δτ₁ ≠ 0, δτ₂ ≠ 0 as $\bar {\rm \xi} _{{\rm final}}$.

Fig. 2.

The final averaged relative distance $\bar {\rm \xi} _{{\rm final}}$ (in the time moment τ_Cfinal = 600, $\bar {\rm \xi} _{{\rm Cfinal}} = 2$) against different phase shifts of pulse peaks δτ₁, δτ₂. Oscillation velocities: η₁ = 3 × 10⁻³, (a) η₂ = 6 × 10⁻², (b) η₂ = 10⁻¹ (field intensities: I ₁ = 3.4 × 10¹² W/cm², (a) I ₂ = 1.3 × 10¹⁵ W/cm², (b) I ₂ = 3.8 × 10¹⁵ W/cm²).

Figure 2a, 2b show dependence of the final distance $\bar {\rm \xi} _{{\rm final}}$ at which the electrons scatter at the time moment τ_Cfinal = 600 for different values of phase shifts δτ₁, δτ₂ and for two values of the oscillation velocity of the second wave. One can see that the final distance $\bar {\rm \xi} _{{\rm final}}$ considerably depends from the oscillation velocity of the second wave and has maximum values for next ranges of the phase shifts: ${\rm \delta} {\rm \tau} _1 \in \left[ { - 600 \div - 50} \right]$, ${\rm \delta} {\rm \tau} _2 \in \left[ {100 \div 500} \right]$. Thus, for the oscillation velocity η₂ = 6 × 10⁻², the final distance can reach the value $\bar {\rm \xi} _{{\rm final}} \approx 160$ (see Fig. 2a), and for the oscillation velocity η₂ = 10⁻¹, the final distance can reach the value $\bar {\rm \xi} _{{\rm final}} \approx 900$ (see Fig. 2b). Figures 3 and 4 show dependence of averaged relative distance $\bar {\rm \xi} $ (in logarithmic units) against the interaction time τ for mainly interesting values of the phase shifts. It is seen that taking into account of the phase shifts of pulse peaks can considerably increase the repulsion force. Thus, for η₁ = 3 × 10⁻³, η₂ = 6 × 10⁻² and δτ₁ = −550,   δτ₂ = 250 ratio $\bar {\rm \xi} _{{\rm final}}/\bar {\rm \xi} _{{\rm Cfinal}} \approx 80$ and ratio $\bar {\rm \xi} _{{\rm final}}/\bar {\rm \xi} ^{(0)}_{{\rm final}} \approx 32$ (see curve 3 in Fig. 3), and for oscillation velocities η₁ = 3 × 10⁻³, η₂ = 10⁻¹ and δτ₁ = −450, δτ₂ = 250 the ratio $\bar {\rm \xi} _{{\rm final}}/\bar {\rm \xi} _{{\rm Cfinal}} \approx 450$ and the ratio $\bar {\rm \xi} _{{\rm final}}/\bar {\rm \xi} ^{(0)}_{{\rm final}} \approx 180$ (see curve 3 in Fig. 4).

Fig. 3.

The averaged relative distance $\bar {\rm \xi} $ (in logarithmic units) against the interaction time τ. The dashed line corresponds to the case of the absence of the external field. The dashed-dot line and solid lines correspond to oscillation velocities: η₁ = 3 × 10⁻³, η₂ = 6 × 10⁻² (field intensities: I ₁ = 3.4 × 10¹² W/cm², I ₂ = 1.3 × 10¹⁵ W/cm²), phase shifts of pulse peaks: 1 – δτ₁ = −450, δτ₂ = 450; 2 – δτ₁ = −350, δτ₂ = 300; 3 – δτ₁ = −550,   δτ₂ = 250; the dashed line with a dot – δτ₁ = 0, δτ₂ = 0.

Fig. 4.

The averaged relative distance $\bar {\rm \xi} $ (in logarithmic units) against the interaction time τ. The dashed line corresponds to case without external field. The dashed-dot line and solid lines correspond to oscillation velocities: η₁ = 3 × 10⁻³, η₂ = 10⁻¹ (the field intensities: I ₁ = 3.4 × 10¹² W/cm², I ₂ = 3.8 × 10¹⁵ W/cm²), the phase shifts of pulse peaks: 1 – δτ₁ = −300, δτ₂ = 350; 2 – δτ₁ = −400, δτ₂ = 300; 3 – δτ₁ = −450, δτ₂ = 250; the dashed-dot line – δτ₁ = 0, δτ₂ = 0.

The effective slowing-down of electrons

The case, when the oscillation velocity η₁ has to be close to the initial relative velocity ${\rm \eta} _1 \approx \dot {\rm \xi} _0$ and the oscillation velocity η₂ is greater an order of magnitude, increasing the interaction time allows us to see oscillations of the effective attraction of electrons. Electrons, after approaching and scattering, get the strong pulse of the attraction and then they re-approach. Let us designate the time at which the averaged relative distance between electrons is equal to $\bar {\rm \xi} _{{\rm Cfinal}} = 2$: in the external field, when δτ₁ = 0,   δτ₂ = 0 – ${\rm \tau} _{{\rm final}}^{(0)} $; in the external field, when δτ₁ ≠ 0,   δτ₂ ≠ 0 – τ_final. Fig. 5 shows dependence of the final distance $\bar {\rm \xi} _{{\rm final}}$ at which electrons scatter at the time τ = 2000 for different values of phase shifts δτ₁, δτ₂ and next values of oscillation velocities η₁ = 1.7 × 10⁻³, η₂ = 3 × 10⁻². It is seen that the final distance $\bar {\rm \xi} _{{\rm final}}$ is smaller $\bar {\rm \xi} _{{\rm Cfinal}} = 2$ and it has the minimum value down to $\bar {\rm \xi} _{{\rm final}} = 10^{ - 1}$ for the next ranges of phase shifts ${\rm \delta} {\rm \tau} _1 \in \left[ {200 \div 400} \right]$, ${\rm \delta} {\rm \tau} _2 \in \left[ {100 \div 600} \right]$.

Fig. 5.

The final averaged relative distance $\bar {\rm \xi} _{{\rm final}}$ (in the time moment τ = 2000, $\bar {\rm \xi} _{{\rm Cfinal}} = 2$) against different phase shifts of the pulse peaks δτ₁, δτ₂. Oscillation velocities: ${\rm \eta} _1 = \dot {\rm \xi} _0 = 1.7 \times 10^{ - 3}$, η₂ = 3 × 10⁻² (field intensities: I ₁ = 1.1 × 10¹² W/cm², I ₂ = 3.4 × 10¹⁴ W/cm²).

Figure 6 shows dependence of averaged relative distance $\bar {\rm \xi} $ on the interaction time τ for mainly interesting values of the phase shifts. One can see that taking into account of the phase shifts can considerably increase the attraction force. Thus, the time of electron scattering to the initial value of the distance ($\bar {\rm \xi} _{{\rm Cfinal}} = 2$) is increased in comparison with the case without an external field to τ_final/τ_Cfinal ≈ 13.5 (see Fig. 6 curve 1 and the dashed line); in the external field when δτ₁ = 0, δτ₂ = 0 the time is increased to ${\rm \tau} _{{\rm final}}/{\rm \tau} _{{\rm final}}^{(0)} \approx 4$ (see Fig. 6 curve 1 and the dashed-dot line) for oscillation velocities η₁ = 1.7 × 10⁻³, η₂ = 3 × 10⁻² and phase shifts δτ₁ = 350, δτ₂ = 400. The effect is a bit weaker for phase shifts δτ₁ = 300, δτ₂ = 300 (see Fig. 6 curve 2).

Fig. 6.

The averaged relative distance $\bar {\rm \xi} $ against the interaction time τ. The dashed line corresponds to case without external field. The dashed-dot line and solid lines correspond to the oscillation velocity: ${\rm \eta} _1 = \dot {\rm \xi} _0 = 1.7 \times 10^{ - 3}$, η₂ = 3 × 10⁻² (the field intensity: I ₁ = 1.1 × 10¹² W/cm², I ₂ = 3.4 × 10¹⁴ W/cm²), phase shifts of pulse peaks: 1 – δτ₁ = 350,   δτ₂ = 400, 2 – δτ₁ = 300,   δτ₂ = 300; the dashed-dot line – δτ₁ = 0,   δτ₂ = 0.