Focusing effects in laser-electron Thomson scattering

We study the effects of laser pulse focusing on the spectral properties of Thomson scattered radiation. Modeling the laser as a paraxial beam we find that, in all but the most extreme cases of focusing, the temporal envelope has a much bigger effect on the spectrum than the focusing itself. For the case of ultrashort pulses, where the paraxial model is no longer valid, we adopt a subcycle vector beam description of the field. It is found that the emission harmonics are blue shifted and broaden out in frequency space as the pulse becomes shorter. Additionally the carrier envelope phase becomes important, resulting in an angular asymmetry in the spectrum. We then use the same model to study the effects of focusing beyond the limit where the paraxial expansion is valid. It is found that fields focussed to subwavelength spot sizes produce spectra that are qualitatively similar to those from subcycle pulses due to the shortening of the pulse with focusing. Finally, we study high-intensity fields and find that, in general, the focusing makes negligible difference to the spectra in the regime of radiation reaction.

Figure 1

Diagram illustrating the coordinate system.

Figure 2

Plot showing the (normalized) amplitude of the longitudinal electric field, $E_z(x,0,z)/a_0$, as a function of the focusing parameter ${\theta }_0$ and the longitudinal coordinate $z$ for different values of $x$. The field component is calculated at $x=w_0/4$ (a), $x=w_0/2$ (b), $x=3w_0/4$ (c) and $x=w_0$ (d). (It should be noted that as we change the focusing parameter the value of $w_0$ also changes.) It can be seen that for weak focusing (${\theta }_0\to 0$) the longitudinal field tends to zero. As we increase the focusing the amplitude rises, becoming a significant fraction of the transverse field.

Figure 3

Plots showing the $\gamma$ factors and velocity components of the electron in a laser pulse with different focusing. The laser pulse has a Gaussian time envelope and was 27 fs in duration with a peak $a_0=2$. Gray (thick) lines: ${\theta }_0=0$. Blue (dashed) lines: ${\theta }_0=0.35$. Red (solid) lines: ${\theta }_0=0.7$.

Figure 4

Thomson emission spectra calculated at ${\theta }_{\mathrm{obs}}=\pi$ for an electron of initial ${\gamma }_0=10$ counterpropagating along the laser axis. In all cases the laser pulse has a peak $a_0=2$ and is 10 cycles in duration (27 fs). It is either a plane wave or paraxial field [focussed to ${\theta }_0=1/\pi$ ($w_0=\lambda$)], with a Gaussian or super Gaussian envelope, as indicated in the panels. The red (dashed) lines show the positions of the odd-numbered harmonics calculated for the case of a monochromatic plane wave.

Figure 5

Illustration showing the electron $\gamma$ factor as it propagates through a focussed pulse with varying impact parameter. In this example the electron had an initial ${\gamma }_0=5$ and was inserted into the laser with impact parameters of $x_i=0$, $x_i=w_0/2$ and $x_i=w_0$, which correspond to the three blue lines in the plot (back to front, respectively). The laser pulse had a peak $a_0=2$, $\lambda =0.8\mu \mathrm{m}$ and was focussed to a waist of $w_0=5\mu \mathrm{m}$ (${\theta }_0=0.05$). The duration was reduced to 20 fs to aid clarity of illustration. The yellow-black shading illustrates the intensity profile at $t=0$. The curved black lines in the $x-z$ plane show the beam waist $w$. The red curve in the $x-\gamma$ plane shows the transverse profile function, $\exp (-r^2/w^2)$, evaluated at $y=z=0$.

Figure 6

Thomson emission spectra for electrons with varying impact parameters $x_i$. The laser is of peak intensity $a_0=2$, $\lambda =0.8\mu \mathrm{m}$, and with a Gaussian profile of 27 fs duration. The incoming electrons have an initial ${\gamma }_0=10$ and are counterpropagating with the laser pulse. Their impact parameters are listed in the figure panels. Black lines: emission spectra calculated for a paraxial laser pulse focussed to ${\theta }_0=1/\pi$, ($w_0=\lambda$). Gray (thick) lines: emission spectra calculated using a plane wave laser, with peak amplitude reduced to compensate for the fall-off in the focussed field (see main text for details). Red (dashed) lines: the positions of the harmonics in the case of a monochromatic field, calculated using the same $a_0$ as the plane wave spectra.

Figure 7

Thomson emission spectra for an electron bunch in a paraxial beam (500 particles randomly distributed over a disk of radius $w_0$ in the transverse plane, shown as blue lines and denoted $N_{\mathrm{para}}=500$) are compared with those of a single particle (yellow lines, denoted $N_{\mathrm{para}}=1$). The electron(s) have initial ${\gamma }_0=10$ and collide head-on with $0.8\mu \mathrm{m}$, 10 cycle laser beam of peak intensity $a_0=2$. Panel (a) shows the results for a focusing of ${\theta }_0=0.85$ ($w_0=0.3\mu \mathrm{m}$), and panel (b) for ${\theta }_0=0.51$ ($w_0=0.5\mu \mathrm{m}$). The spectra is also calculated for the bunch using a plane wave field multiplied by a transverse envelope function $\exp (-r^2/w_0^2)$ (red lines, denoted $N_{\text{plane}}=500$).

Figure 8

Diagram showing the location of the singularity in the vector beam model in relation to the particle orbit.

Figure 9

Emission spectra for laser pulses of decreasing duration (FWHM). The laser is of peak intensity $a_0=2$ and wavelength $\lambda =0.8\mu \mathrm{m}$. The electron has an initial ${\gamma }_0=10$. Gray (thick) lines: spectra calculated for a plane wave. Black (solid) lines: spectra calculated using a 5th order paraxial beam focussed to a spot size of $w_0=0.8\mu \mathrm{m}$ (${\theta }_0=1/\pi$). Blue (dash-dot) lines: the same but modeled using the vector beam model. Red (dashed) lines: the locations of the harmonic peaks for the case of an infinite plane wave (see Sec. 3a).

Figure 10

Electron motion and corresponding emission spectra in short laser pulses. The laser is described using the vector beam model and has a peak $a_0=2$, $\lambda =0.8\mu \mathrm{m}$ and is focussed to a spot size of $w_0=0.8\mu \mathrm{m}$ (${\theta }_0=1/\pi$). The electron collides along the propagation axis and has an initial ${\gamma }_0=10$. Top row: results for a 2 cycle (FWHM) pulse. Bottom row: results for a 0.5 cycle (FWHM) pulse. Left column: transverse velocity, $u_x$, and transverse trajectory $10\times k·x$ (units dimensionless). Center column: emission spectra as a function of frequency and angle. Right column: angular emission spectra integrated over all frequency (arb. units). The dashed line indicates the propagation direction ($\theta =\pi$). (All units dimensionless.)

Figure 11

Plots showing the total radiated power (Larmor power) in dimensionless units for fixed $a_0$ (a,b,c) and for fixed ${\gamma }_0$ (d,e,f) using the vector beam model (VBM) and the 1st and 5th order paraxial beam model (PBM). The relative $\%$ error in radiated power for the paraxial beam with respect to the vector beam model is presented in (g) and (h). The lines denoted by ($a_1$) and ($f_1$) are for the case of the 1st order paraxial beam, in all other cases there are no visible differences between the 1st and 5th order models.

Figure 12

Spectra and trajectories for an electron of initial ${\gamma }_0=10$ colliding with a laser of peak $a_0=2$, duration 10 cycles (27 fs) and wavelength $\lambda =0.8\mu \mathrm{m}$, for different levels of focusing. Left panels show emission spectra, right panels show the trajectories. Top row (blue lines): plane wave. Middle row (blue lines): vector beam focussed to ${\theta }_0=0.8$ ($w_0=0.318\mu \mathrm{m}$). Bottom row (blue lines): vector beam focussed to ${\theta }_0=1.2$ ($w_0=0.212\mu \mathrm{m}$). The red lines show the same but are calculated using a plane wave with duration shortened to match the actual FWHM duration of the focussed field (which becomes shortened by the focusing). For the middle panels this duration is 3.5 cycles, and for the bottom panels 2.5 cycles.

Figure 13

Emission spectra for an electron of initial ${\gamma }_0=10$ in collision with a 10 cycle FWHM laser of peak $a_0=2$ and $\lambda =0.8\mu \mathrm{m}$. (a) Laser is a plane wave. (b) Laser is modeled as a 5th order paraxial beam focussed to a spot size of $w_0=0.8\mu \mathrm{m}$ (${\theta }_0=1/\pi$). (c) a 5th order paraxial beam focussed to a spot size of $w_0=0.2\mu \mathrm{m}$ (${\theta }_0=1.2$). (d) The same as (c) but calculated using the vector beam model.

Figure 14

Angular emission spectra integrated over all frequencies for fields of different focusing. In all cases the laser has a peak $a_0=2$, $\lambda =0.8\mu \mathrm{m}$ and has an envelope function that is 10 cycles in duration. The incoming electron has an initial ${\gamma }_0=10$.

Figure 15

Electron dynamics for three different focusing parameters, with RR included. These are $w_0=0.3\mu \mathrm{m}$ (${\theta }_0=0.85$), $w_0=0.5\mu \mathrm{m}$ (${\theta }_0=0.51$), and $w_0=1.0\mu \mathrm{m}$ (${\theta }_0=0.25$). In all cases the laser is modeled as a paraxial beam of peak intensity $a_0=200$, wavelength $0.8\mu \mathrm{m}$ and duration 27 fs. The electron has an initial ${\gamma }_0=500$ and is injected along the laser axis. The right hand panel shows the longitudinal electric field (normalized to $e{E}_z/{\omega }_{0}m$) as experienced by the particle.

Figure 16

Electron dynamics for three different impact parameters, with RR included. In all cases the laser is modeled as a paraxial beam of peak intensity $a_0=200$, wavelength $0.8\mu \mathrm{m}$ and duration 27 fs. The electron has an initial ${\gamma }_0=500$ and is injected along the laser axis. The two lower panels show the longitudinal electric field (normalized to $e{E}_z/{\omega }_{0}m$) as experienced by the particles.

Figure 17

Thomson emission spectra calculated in the backscattering direction, ${\theta }_{\mathrm{obs}}=\pi$, for the electrons in Fig. 15. Panel (a) shows the spectra for the exact setup described in Fig. 15, while panel (b) shows the spectra calculated using a circularly polarized laser pulse with the same peak intensity and duration.