Dephasingless plasma wakefield photon acceleration

Sandberg and Thomas [Phys. Rev. Lett. 130, 085001 (2023)] proposed a scheme to generate ultrashort, high-energy pulses of XUV photons though dephasingless photon acceleration in a beam-driven plasma wakefield. An ultrashort laser pulse is placed in the plasma wake behind a relativistic electron bunch such that it experiences a comoving negative density gradient and therefore shifts up in frequency. Using a tapered density profile provides phase-matching between driver and witness pulses. In this paper, we give the details of the wakefield solutions and phase-matching conditions used to generate the phase-matching density profile. The short, high-density, and weak driver limits are considered. We show, explicitly, the numerical algorithm used to calculate the density profiles.

Figure 1

Dependence of the parameter $x_d$ on the drive-beam parameters $k_p{L}_d$ and $n_d/n$. The dashed red line indicates the trajectory a dePWPA solution starting at $(k_p{L}_d,n_d/n)=(20,0.0375)$ takes in ($k_p{L}_d, n_d/n$). The solution is parameterized by the plasma density $n:1\to 0$. Note the constancy of $x_d$ with $n$ as $k_p{L}_d$ gets small or $n_d/n$ gets large; this is a prediction of the short or high-density (SHD) approximation discussed in Sec. 5.

Figure 2

Numerically determined profiles for unlimited photon acceleration, computed by Algorithm lg1 with $n_d=0.4, L_d=1$, and $A=0.4$. (a) shows the computed plasma density and (b) shows the relative gain in laser frequency.

Figure 3

Comparing numerical and short or high-density small $A$ expansions. We show that the wake amplitude ${\gamma }_m$, the position of 0 density perturbation ${\zeta }_{\delta }$, and the predicted density $n$ and frequency gain $k_L/k_{L0}$ profiles for $A=0.01,0.1,1$, and 10 when $k_{p0}{L}_{d0}=0.01$. The length satisfies $k_{p0}{L}_d\ll 1$, so ${\gamma }_m=1+A^2/2n$ is always accurate. The expansion to $\mathcal{O}\left(A^2\right)$ is quite accurate for ${\zeta }_{\delta },n$, and $k_L$ for $A$ as large as 1. The expansion when $A=10$ is not good for ${\zeta }_{\delta }$ but is surprisingly accurate in predicting $n$ and $k_L$ even for $A=10$.

Figure 4

Analytic expressions in the wake compared to OSIRIS 1D simulation data. Top row: $k_{p0}{L}_{d0}=1.9,n_{d0}/n=0.2,0.5,0.8$. Middle row: $k_{p0}{L}_{d0}=6.28,n_{d0}/n=0.1,0.5,0.6$ Bottom row: $k_{p0}{L}_{d0}=9.5,n_{d0}/n=0.1,0.5,0.6.$ The expressions are exact in a uniform density plasma (here $n=1$ but any value of $n$ is possible).

Figure 5

Jupyter notebook output demonstrating usage of the Python code dephasingless_pwpa_plasma_profile to calculate the dephasingless PWPA plasma density profile and theoretical laser frequency.

Figure 6

A sample help message for the get_depa_profile function in the dephasingless_pwpa_plasma_profile Python code.