Frequency scaling law for nonlinear Compton and Thomson scattering: Relevance of spin and polarization effects

The distributions of Compton and Thomson radiation for a shaped laser pulse colliding with a free electron are calculated in the framework of quantum and classical electrodynamics, respectively. We introduce a scaling law for the Compton and the Thomson frequency distributions which universally applies to long and short incident pulses. Thus, we extend the validity of frequency scaling postulated in previous studies comparing nonlinear Compton and Thomson processes. The scaling law introduced in this paper relates the Compton no-spin flipping process to the Thomson process over nearly the entire spectrum of emitted radiation, including its high-energy portion. By applying the frequency scaling, we identify that both spin and polarization effects are responsible for differences between classical and quantum results. The same frequency scaling applies to angular distributions and to temporal power distributions of emitted radiation, which we illustrate numerically.

Figure 1

Energy spectra for the Compton scattering (solid blue line) [Eq. (13)] and for the Thomson scattering (dashed red line) [Eq. (20)] for the linearly polarized laser field propagating in the $z$ direction with the polarization vector along the $x$ axis. The laser pulse parameters are $\mu =1$, $N_{\mathrm{osc}}=32$, and ${\omega }_{\mathrm{L}}=3\times {10}^{-6}{m}_{\mathrm{e}}{c}^2$. The scattered radiation is linearly polarized in the scattering plane, and it is characterized by the polar and azimuthal angles ${\theta }_{\mathit{K}}=0.98\pi$ and ${\phi }_{\mathit{K}}=0$, respectively. The initial electron propagates in the opposite direction with respect to the $z$ axis, with momentum $|{\mathit{p}}_{\mathrm{i}}|=50m_{\mathrm{e}}c$. In the upper panel, the Thomson spectrum is calculated for the frequency ${\omega }_{\mathit{K}}^{\mathrm{Th}}$. In the lower panel, this frequency is transformed to the Compton frequency ${\omega }_{\mathit{K}}$ by applying the scaling law (33) and, in addition, the Thomson energy spectrum is multiplied by 2. For these particular parameters, ${\omega }_{\mathrm{cut}}\approx 50m_{\mathrm{e}}{c}^2$.

Figure 2

Energy spectra for the Compton scattering (solid blue line for the no-spin-flipping process ${\lambda }_{\mathrm{i}}{\lambda }_{\mathrm{f}}=1$) [Eq. (14)] and for the Thomson scattering (dashed magenta and solid red lines) [Eq. (20)]. The driving pulse propagates in the $z$ direction and is linearly polarized along the $x$ axis. The remaining laser field parameters are such that $\mu =10$, $N_{\mathrm{osc}}=16$, and ${\omega }_{\mathrm{L}}=0.3m_{\mathrm{e}}{c}^2$. The direction of scattered radiation is given by the polar and azimuthal angles ${\theta }_{\mathit{K}}=0.99\pi$ and ${\phi }_{\mathit{K}}=0$, respectively. These parameters are specified in the rest frame of incident electrons. The Thomson spectrum, multiplied by the factor 0.9, is calculated for the frequency ${\omega }_{\mathit{K}}^{\mathrm{Th}}$. In this reference frame and for these parameters, ${\omega }_{\mathrm{cut}}\approx m_{\mathrm{e}}{c}^2/2$.

Figure 3

Energy spectra for the Compton scattering (solid blue line for the no-spin-flipping process ${\lambda }_{\mathrm{i}}{\lambda }_{\mathrm{f}}=1$, dashed magenta line for the spin-flipping process ${\lambda }_{\mathrm{i}}{\lambda }_{\mathrm{f}}=-1$) [Eq. (14)], and for the Thomson scattering (solid red line, reflected with respect to the horizontal black line) [Eq. (20)], and for the same parameters as in Fig. 2. In the left column, the energy spectra are presented for emitted radiation polarized linearly in the scattering plane for three chosen frequency domains. The right column displays the energy spectra for perpendicularly polarized emitted radiation, for which the Thomson theory gives 0. The Thomson spectrum is calculated for the frequency ${\omega }_{\mathit{K}}^{\mathrm{Th}}$ and then the frequency is transformed to the Compton frequency ${\omega }_{\mathit{K}}$ by applying the scaling law (33). For these particular parameters and for the reference frame considered, ${\omega }_{\mathrm{cut}}\approx m_{\mathrm{e}}{c}^2/2$.

Figure 4

Energy spectra for the Compton scattering (the solid blue line is for the no-spin-flipping process ${\lambda }_{\mathrm{i}}{\lambda }_{\mathrm{f}}=1$, the solid magenta (light gray) line is for the spin-flipping process ${\lambda }_{\mathrm{i}}{\lambda }_{\mathrm{f}}=-1$) [Eq. (14)] and for the Thomson scattering (red line, reflected with respect to the horizontal black line) [Eq. (20)]. The presented results are for the laser pulse propagating in the $z$ direction with a linear polarization vector along the $x$ axis. The remaining parameters are $\mu =10$, $N_{\mathrm{osc}}=2$, and ${\omega }_{\mathrm{L}}=0.3m_{\mathrm{e}}{c}^2$. The direction of scattered radiation is given by the polar and azimuthal angles ${\theta }_{\mathit{K}}=0.99\pi$ and ${\phi }_{\mathit{K}}=0$. These parameters are in the reference frame of incident electrons. In the upper panel, the energy spectra are presented for radiation emitted with a linear polarization in the scattering plane. In the lower panel, the energy spectra of Compton radiation polarized perpendicularly to the scattering plane are displayed; note that in this case the Thomson theory gives 0. The Thomson spectrum is calculated for the frequency ${\omega }_{\mathit{K}}^{\mathrm{Th}}$ and then the frequency is transformed to the Compton frequency ${\omega }_{\mathit{K}}$ by applying the scaling law (33). For these particular parameters and for the chosen reference frame, ${\omega }_{\mathrm{cut}}\approx m_{\mathrm{e}}{c}^2/2$.

Figure 5

The left column represents the same as in Fig. 4, but with ${\theta }_{\mathit{K}}=0.5\pi$, for which ${\omega }_{\mathrm{cut}}=m_{\mathrm{e}}{c}^2$. The right column shows two enlarged parts of the upper-left frame, but only for the no-spin-flipping processes. We observe very good agreement between the Compton and Thomson results (up to the multiplicative factor) even for ${\omega }_{\mathit{K}}/{\omega }_{\mathrm{cut}}$ close to 1.

Figure 6

The same as in Figs. 2 and 3, but for ${\omega }_{\mathit{K}}$ very close to the cutoff value ${\omega }_{\mathrm{cut}}$. In the upper panel, the Compton (for the spin-conserved process and with the Compton photon linearly polarized in the scattering plane) and Thomson (mirror-reflected) energy spectra are compared such that the Thomson distribution is normalized to the maximum value of the Compton one. Although in this frequency domain both distributions exhibit rather irregular behavior, after frequency transformation and normalization there is the perfect agreement between quantum spin-conserved and classical theories. In the lower panel, the Compton distribution from the upper frame (dashed line) is compared to the total-energy distribution for the Compton process (solid line), when summed over all final spin and polarization degrees of freedom, and averaged over the initial spins. In this frequency domain, all spin and polarization degrees of freedom contribute significantly to the total distribution.

Figure 7

The same as in Fig. 3, but only for the no-spin-flipping process (${\lambda }_{\mathrm{i}}{\lambda }_{\mathrm{f}}=1$), fixed frequency of generated radiation, ${\omega }_{\mathit{K}}=0.2024m_{\mathrm{e}}{c}^2$, ${\Phi }_{\mathit{K}}=0$, and for two domains of the emission angle ${\Theta }_{\mathit{K}}$. In both cases, the polar-angle distributions for the Compton (blue line) and Thomson (mirrored red line) scattering show the structural similarity as for the frequency distributions.

Figure 8

Azimuthal-angle distribution of radiation energy generated by the no-spin-flipping Compton process (blue line) and by the frequency-scaled Thomson scattering (mirrored red line) for the same laser pulse parameters as in Figs. 4 and 5. The upper row presents distributions of radiation linearly polarized in the laser pulse plane [cf. Eq. (47)], and the lower row for polarization (48). In the left column, we show the results for ${\omega }_{\mathit{K}}=0.2024m_{\mathrm{e}}{c}^2$, and in the right one for ${\omega }_{\mathit{K}}=0.6m_{\mathrm{e}}{c}^2$. For all cases, the polar angle ${\theta }_{\mathit{K}}=0.7\pi$. These distributions satisfy the symmetry ${\phi }_{\mathit{K}}\to 2\pi -{\phi }_{\mathit{K}}$.

Figure 9

Total energy (in relativistic units) in the electron's reference frame for linearly polarized laser pulse defined by Eq. (8), ${\omega }_{\mathrm{L}}=0.3m_{\mathrm{e}}{c}^2$ and $N_{\mathrm{osc}}=16$. In the log-log plot we present the total energy calculated for the Compton process (dark red circles) and for the Thomson process (light brown diamonds). The continuous blue (dark) and green (light) lines are to guide the eye. The thin black straight line represents the fitting curve defined by Eq. (60).

Figure 10

The same as in Fig. 9 but for $N_{\mathrm{osc}}=2$.

Figure 11

Temporal power distribution of electromagnetic radiation generated by Compton and Thomson scattering, and synthesize from energy distributions presented in Fig. 6. In the upper panel, we compare temporal power distributions for generated radiation linearly polarized in the scattering plane for the spin-conserved Compton process (blue line) and the frequency-scaled Thomson process (mirrored red line). Both distributions are normalized to their maximum values. In the lower panel, we compare the temporal power distribution for Compton scattering from the upper panel (dashed red line) with the total power distribution summed over the final spin and polarization degrees of freedom and averaged over the initial spins (solid blue line). We see that spin and polarization effects observed for power distributions are even more pronounced than the ones observed for frequency distributions.