Interference effects in nonlinear Compton scattering due to pulse envelope

Nonlinear Compton scattering is calculated for the collision of an electron with a plane-wave pulse. A midinfrared (IR) peak arises in the photon spectrum due to long-range interference associated with the pulse envelope. The case of a flattop pulse is studied as a toy model for pulse envelope effects and reduced to two final-state momentum integrations; the case of a sine-squared pulse is studied numerically. A perturbative expansion in charge-field coupling reveals that already at intermediate intensities, many orders are required to correctly capture the structure of the mid-IR peak. By regularizing the classical result, it is shown that the mid-IR peak is due to plane-wave ponderomotive effects from the pulse envelope. Finally, it is shown that the mid-IR peak can be isolated using energy, angle and polarization filters.

Figure 1

(a),(b) Compton light front momentum spectra ($d\mathcal{I}/ds$) for the flattop pulse with $n_{*}=0$. The amplitude has been normalized to unit IR limit (${\xi }^2\mathrm{\Phi }/2$). $N$ is the number of laser cycles [$N=4$ in (a)]. In (c), the error in the normalized spectrum is plotted, for when a different lowest harmonic, $n_{*}$, is chosen for the harmonic sum: $\mathcal{I}(n_{*}=-1)/\mathcal{I}(n_{*})-1$.

Figure 2

How the mid-IR peak emerges as $\xi$ is increased. The horizontal axis on both figures is identical. The curves in the lower plot correspond to the lineouts indicated by solid lines on the upper plot. The dashed line is the approximate position of the mid-IR peak, which is shown on the lineouts by a dot on each curve. The results are for an $\eta =0.1$ electron colliding head on with a four-cycle ($N=4$) pulse.

Figure 3

The relative height of the mid-IR peak compared to the IR limit ($\propto {\xi }^2$) grows linearly ($\propto \xi$) when $\xi \gtrsim 1$.

Figure 4

Perturbative expansion of the photon spectrum for a four-cycle, $\xi =0.5$ pulse colliding with an $\eta =0.1$ electron. $j^{\max }$ corresponds to the highest order included in $\xi$ from a truncated perturbation expansion of the amplitude. (Shading and plot points have been used to guide the eye to where the perturbation expansion breaks down.) $j^{\max }=1$ corresponds to linear Compton scattering. The higher the maximum perturbative order included, the higher the value of light front momentum $s$, until which the expansion remains accurate. The vertical grid lines are the position of the nonlinear (left) and linear (right) Compton edge.

Figure 5

Perturbative expansion of the photon spectrum for a $\xi =2.5$, 16-cycle pulse colliding with an $\eta =0.1$ electron; cf. Fig. 4. (Shading and plot points have been used to guide the eye to where the perturbation expansion breaks down.) The vertical grid lines are the estimate of the position of the mid-IR peak (left) and the first harmonic (right).

Figure 6

Comparison of different contributions to classical radiation spectrum ($\xi =2.5$, $\eta =0.1$, $N=4$) by removing different parts of the squared momentum in (14). $K_{\perp +0}^{\mathrm{cl}}$ corresponds to removing the longitudinal “$\parallel$ term” in (14), whereas $K_{\perp }^{\mathrm{cl}}$ is for where the longitudinal and zero-field terms having been removed. The LMA, which misses the mid-IR peak, is plotted for comparison. The vertical grid lines are the harmonics and the approximation to the position of the mid-IR bump.

Figure 7

A plot of the square root of the emission spectrum in the transverse, ${\mathbf{r}}^{\perp }$, plane, for a 16-cycle $\xi =2.5$ pulse colliding with an $\eta =0.1$ electron producing $s=5\times {10}^{-4}$ photons (for a head-on collision with 1.55 eV laser photons, this corresponds to 4 MeV Compton-scattered photons). The outer ring is part of the first harmonic, whereas the inner lobes are the linearly polarized signal from the mid-IR peak. Dashed lines are plotted at angles $\pm \pi /4$ to the vertical.

Figure 8

Angular spectrum for the collision of a 16-cycle, $\xi =2.5$ pulse with an $\eta =0.1$ electron. Left: how the angular spectrum depends on photon polarization (same scale). Right upper: a lineout of the angular spectrum, $L$, at $r^{\perp }=\xi$. Right lower: proportion of $s$ spectrum of photons originating from mid-IR peak $L^{\mathrm{IR}}/L$ (defined as $s<0.01$) as a function of $r^{\perp }/\xi$.