Lorentz-Abraham-Dirac and Landau-Lifshitz equations of motion and the solution to a relativistic electron in a counterpropagating laser beam

Beginning with a critical examination of the Lorentz-Abraham (LA) classical equation of motion for an extended charge and the closely related Lorentz-Abraham-Dirac (LAD) equation of motion for a mass-renormalized point-charge, the Landau-Lifshitz (LL) approximate solution to the LAD equation of motion is determined for an electron subject to a counterpropagating linearly or circularly polarized plane-wave pulse with an arbitrarily shaped envelope. A convenient three-vector formulation of the LL equation is used to derive closed-form expressions for the velocities and associated powers of the electron directly in terms of the time in the laboratory frame. The three-vector formulation also reveals definitive criteria for the LL solution to be an accurate approximation to the LAD equation of motion and for the LL solution to reduce to the solution of the Lorentz force equation of motion that ignores radiation reaction. Semiclassical analyses are used to obtain simple conditions for determining the regimes where the quantum effects of either Compton electron scattering by the incident photons or electron recoil produced by the emitted photons is significant. It is proven that the LL approximation becomes an inaccurate solution to the LAD equation of motion only for large enough electron velocities and plane-wave intensities that quantum recoil effects on the electron can greatly alter the classical solution. Comparisons are made with previously published analytical and numerical solutions to the LL equation of motion for the velocity of an electron in a counterpropagating plane wave.

Figure 1

$P_{\mathrm{ext}}$, $P_{\mathrm{rad}}$, and $P_{\mathrm{kin}}$ plotted vs time $t$ for a uniform linearly polarized laser with angular frequency $\omega =2\times {10}^{15}{\mathrm{s}}^{-1}$ and strength $a_0=100$ ($\mathcal{E}=1$), and an electron with initial relativistic factor ${\gamma }_0=1000$.

Figure 2

Comparison of $P_{\mathrm{rad}}(t)$ and $P_{\mathrm{rad}}^{\mathrm{LAD}}(t)$ for the uniform linearly polarized plane wave with parameters given in Fig. 1.

Figure 3

Comparison of $P_{\text{Sch}}(t)$ and $P_{\text{Sch}}^{\mathrm{LAD}}(t)$ for the uniform linearly polarized plane wave with parameters given in Fig. 1.

Figure 4

Relativistic factor $\gamma$ plotted vs time $t$ for a sinusoidal-envelope linearly polarized laser pulse with angular frequency $\omega =2\times {10}^{15}{\mathrm{s}}^{-1}$, maximum strength $a_0=100$, and a pulse-width equal to 26.8 fs. The electron has an initial relativistic factor ${\gamma }_0=1000$.

Figure 5

Longitudinal velocity $u_z/c$ plotted vs time $t$ for a sinusoidal-envelope linearly polarized laser pulse with parameters given in Fig. 4.

Figure 6

Transverse velocity $u_x/c$ plotted vs time $t$ for a sinusoidal-envelope linearly polarized laser pulse with parameters given in Fig. 4.

Figure 7

$P_{\mathrm{ext}}$, $P_{\mathrm{rad}}$, and $P_{\mathrm{kin}}$ plotted vs time $t$ for a sinusoidal-envelope linearly polarized laser with parameters given in Fig. 4.

Figure 8

Comparison of $P_{\mathrm{rad}}(t)$ and $P_{\mathrm{rad}}^{\mathrm{LAD}}(t)$ for the sinusoidal-envelope linearly polarized plane wave with parameters given in Fig. 4.

Figure 9

Comparison of $P_{\text{Sch}}(t)$ and $P_{\text{Sch}}^{\mathrm{LAD}}(t)$ for the sinusoidal-envelope linearly polarized plane wave with parameters given in Fig. 4.

Figure 10

$P_{\mathrm{ext}}$, $P_{\mathrm{rad}}$, and $P_{\mathrm{kin}}$ plotted vs time $t$ for a uniform circularly polarized laser with angular frequency $\omega =2\times {10}^{15}{\mathrm{s}}^{-1}$ and strength $a_0=100$ ($\mathcal{E}=1$), and an electron with initial relativistic factor ${\gamma }_0=1000$.

Figure 11

$P_{\text{Sch}}(t)$ and $P_{\text{Sch}}^{\mathrm{LAD}}(t)$ for the uniform circularly polarized plane wave with parameters given in Fig. 10.

Figure 12

Relativistic factor $\gamma$ plotted vs time $t$ for a sinusoidal-envelope circularly polarized laser pulse with angular frequency $\omega =2\times {10}^{15}{\mathrm{s}}^{-1}$, maximum strength $a_0=100$, and a pulse-width equal to 26.8 fs. The electron has an initial relativistic factor ${\gamma }_0=1000$.

Figure 13

Longitudinal velocity $u_z/c$ plotted vs time $t$ for a sinusoidal-envelope circularly polarized laser pulse with parameters given in Fig. 12.

Figure 14

Transverse velocity $u_x/c$ plotted vs time $t$ for a sinusoidal-envelope circularly polarized laser pulse with parameters given in Fig. 12.

Figure 15

$P_{\mathrm{ext}}$, $P_{\mathrm{rad}}$, and $P_{\mathrm{kin}}$ plotted vs time $t$ for a sinusoidal-envelope circularly polarized laser with parameters given in Fig. 12.

Figure 16

$P_{\text{Sch}}(t)$ and $P_{\text{Sch}}^{\mathrm{LAD}}(t)$ for the sinusoidal-envelope circularly polarized plane wave with parameters given in Fig. 12.

Figure 17

The ${\log }_{10}(\gamma )$ vs ${\log }_{10}(a_0)$ plot of the regions of validity for the relativistic electron in a counterpropagating laser beam with $\omega =2\times {10}^{15}{\mathrm{s}}^{-1}$ and $u_z/c\approx 1$. Above the solid red line, quantum recoil effects of the emitted photons may be significant. Above the dotted black line, quantum Compton scattering of the individual incident photons may be significant. Above the dashed blue line, the LL solution may not be an accurate approximation to the LAD solution.