Towards ultimate bandwidth photon sources based on Compton backscattering: Design constraints due to nonlinear effects

Nuclear resonance fluorescence experiments typically require high rates of monochromatic photons due to the narrow linewidth of these resonances. Inverse Compton scattering sources are used to perform these experiments. Their intrinsic excellent monochromaticity is however spoiled by a variety of unavoidable imperfections related to the electron and laser beams. Some projects aim at reaching one per-mille of energy bandwidth, which requires attaining excellent brilliance of the electron beam but also a careful optimization of the laser-beam parameters. In particular, in such a situation, a careful accounting for the nonlinearities induced by a relatively large laser energy has to be considered. In this article, we revisit these nonlinearities with a quantum viewpoint with the goal to provide analytical expressions that can be employed for a very fast optimization of the performance of the source. These expressions were benchmarked against the CAIN event generator with an excellent accuracy in the parameters hypervolume that is of interest in this context. We also show that previously published expression often used to include laser nonlinearities in analytical bandwidth expressions significantly depart from the detailed CAIN simulations. The obtained expression are further used to optimize designs similar to those considered in on-going projects.

Figure 1

Average energy (top) and relative bandwidth (bottom) of the photon beam estimated with CAIN (black dots) and using the analytical formula (red line) given in Eqs. (26) and (27) as a function of the angular aperture. The statistical uncertainty related to the finite statistics of the CAIN simulation is smaller that the size of the dots. The two curves are superimposed within a few ${10}^{-4}$.

Figure 2

Average energy (top) and relative bandwidth (bottom) computed using CAIN (black dots) and using the analytical formula (red line) given for plural scattering combined with effect related to electron dressing as a function of the angular aperture. The two curves are superimposed for most values of apertures.

Figure 3

Photon energy spectrum with all nonlinearities included for a circularly (linearly) polarized laser beam in red (black) in log-scale (top) and zoomed around the first order contribution within $200\mu \mathrm{rad}$ aperture (bottom). On the latter the two distribution are hardly distinguishable by eye, as expected.

Figure 4

The fraction of photons for the second (quadratic curves) and third (nearly linear curves) harmonics for purely circularly (red line and black diamonds) and linearly (blue lines and black dots and squared) polarized laser beams using the analytical formula (blue and red solid lines), and CAIN simulations (black markers).

Figure 5

Relative bandwidth of the photon beam estimated with CAIN (black dots), with the model proposed in this article (open squares) and the model of Ref. [11] (black crosses) as function of ${\overline{\eta }}^2$. Lines are drawn as a further guide to the eye for the analytical models. The laser pulse energy is varied from 1 to 10 J to realize this study. An aperture of $300\mu \mathrm{rad}$ has been used to compute the relative bandwidth.

Figure 6

Relative bandwidth of the photon beam estimated with CAIN (black dots), with the model proposed in this article (open red squares, red line), the model of Ref. [11] (black crosses, black line) and the same without the nonlinear term (black diamonds, dashed line) as function of $X_0$. Lines are drawn as a further guide to the eye for the analytical models. The electron beam energy is varied from 300 to 900 MeV to realize this study. In order to be insensitive to the scaling of aperture ${\theta }_{\max }$ with energy, ${\mathrm{\Psi }}_0$ is kept constant by choosing ${\theta }_{\max }=300\mu \mathrm{rad}$, $150\mu \mathrm{rad}$, and $100\mu \mathrm{rad}$ for 300 MeV, 600 MeV, and 900 MeV, respectively.

Figure 7

The minimal achievable bandwidth (for a vanishing aperture) as function of the laser beam pulse duration, for a transverse size of ${\sigma }_{x,y;l}=7.5\mu \mathrm{m}$ in both horizontal and vertical directions and laser beam single pulse energies of 25 mJ (black dots), 100 mJ (red squares), and 400 mJ (blue diamonds). The similar curve with formula of Ref. [11] for a 25 mJ laser beam is given with pink crosses.

Figure 8

The spectral density vs the ICS bandwidth for a laser pulse energy of $U=400\mathrm{mJ}$, (black dots) ${\sigma }_{t;l}=0.5\mathrm{ps}$, and ${\sigma }_{x;l}={\sigma }_{y;l}=7.5\mu \mathrm{m}$, (gray squares) ${\sigma }_{t;l}=1.25\mathrm{ps}$ and ${\sigma }_{x;l}={\sigma }_{y;l}=7.5\mu \mathrm{m}$, (black triangles) ${\sigma }_{t;l}=3\mathrm{ps}$ and ${\sigma }_{x;l}={\sigma }_{y;l}=7.5\mu \mathrm{m}$, and (gray crosses) ${\sigma }_{t;l}=1.25\mathrm{ps}$ and ${\sigma }_{x;l}={\sigma }_{y;l}=12.5\mu \mathrm{m}$.

Figure 9

The spectral density vs the ICS bandwidth for a laser pulse energy of $U=25\mathrm{mJ}$, (black dots) ${\sigma }_{t,l}=0.5\mathrm{ps}$ and ${\sigma }_{x,l}={\sigma }_{y,l}=7.5\mu \mathrm{m}$, (gray squares) ${\sigma }_{t,l}=1.25\mathrm{ps}$ and ${\sigma }_{x,l}={\sigma }_{y,l}=7.5\mu \mathrm{m}$, (black triangles) ${\sigma }_{t,l}=3\mathrm{ps}$ and ${\sigma }_{x,l}={\sigma }_{y,l}=7.5\mu \mathrm{m}$, and (gray crosses) ${\sigma }_{t,l}=1.25\mathrm{ps}$ and ${\sigma }_{x,l}={\sigma }_{y,l}=12.5\mu \mathrm{m}$.

Figure 10

The fraction of photons emitted on the second harmonic (black increasing quadratic curves with aperture) and third harmonic (gray linear decreasing curves with aperture) for a laser beam with pure circular (full and open squares) or linear (full and open dots or crosses) polarization as function of aperture. They are shown for a $U=400\mathrm{mJ}$ pulse of ${\sigma }_{t,l}=1.25\mathrm{ps}$ and ${\sigma }_{x,l}=7.5\mu \mathrm{m}$ (full markers), and ${\sigma }_{x,l}=12.5\mu \mathrm{m}$ (open markers). The third harmonic contribution is negligible in the ${\sigma }_{x,l}=12.5\mu \mathrm{m}$ case.

Figure 11

The spectral density vs the ERL ICS bandwidth for a laser pulse energy of $U=25\mathrm{mJ}$, (black dots) ${\sigma }_{t,l}=0.5\mathrm{ps}$ and ${\sigma }_{x,l}={\sigma }_{y,l}=7.5\mu \mathrm{m}$, (gray squares) ${\sigma }_{t,l}=1.25\mathrm{ps}$ and ${\sigma }_{x,l}={\sigma }_{y,l}=7.5\mu \mathrm{m}$, (black triangles) ${\sigma }_{t,l}=3\mathrm{ps}$ and ${\sigma }_{x,l}={\sigma }_{y,l}=7.5\mu \mathrm{m}$, and (gray crosses) ${\sigma }_{t,l}=1.25\mathrm{ps}$ and ${\sigma }_{x,l}={\sigma }_{y,l}=12.5\mu \mathrm{m}$.