Tunable polarization plasma channel undulator for narrow bandwidth photon emission

The theory of a plasma undulator excited by a short intense laser pulse in a parabolic plasma channel is presented. The undulator fields are generated either by the laser pulse incident off-axis and/or under the angle with respect to the channel axis. Linear plasma theory is used to derive the wakefield structure. It is shown that the electrons injected into the plasma wakefields experience betatron motion and undulator oscillations. Optimal electron beam injection conditions are derived for minimizing the amplitude of the betatron motion, producing narrow-bandwidth undulator radiation. Polarization control is readily achieved by varying the laser pulse injection conditions.

Figure 1

Laser pulse centroid $⟨x⟩(M_p/2\pi )$ (dashed line) and laser spotsize $w(M_p/2\pi )$ (dotted and solid lines) as functions of the propagation distance $z^{\prime }/{\lambda }_L$. Initial centroid displacement ${⟨x⟩}_0(M_p/2\pi )=1$. The channel parameters are chosen in such a way that the matched laser pulse spotsize is equal to $r_m(M_p/2\pi )=7$. Matched (solid line), with $w_0=r_m$, and mismatched (dotted line), with $w_0=1.5r_m$ cases are shown.

Figure 2

Electron dimensionless transverse momenta components $u_x$ and $u_y$ as functions of the propagation distance $z^{\prime }/{\lambda }_L$. The blue curve corresponds to the case of injection of an electron with $(x_{e,0},y_{e,0})=(-a_u/({\gamma }_0\mathrm{\Omega }),0)$ and $(u_{x,0},u_{y,0})=(0,-a_u)$ into the circular undulator with $a_u=1$. (a) The red curve is the case of the electron injection with $(x_{e,0},y_{e,0})=(0,0)$ and $(u_{x,0},u_{y,0})=(0,0)$. (b) The green line corresponds to the case of the electron injection conditions with $(x_{e,0},y_{e,0})=(-a_u/({\gamma }_0\mathrm{\Omega }),0)$ and $(u_{x,0},u_{y,0})=(0,-a_u)$, for the case when the exponential term in Eq. (18) is included.

Figure 3

Electron beam evolution inside the plasma undulator. The color image at each position $z^{\prime }$ represents the distribution of the $x^{\prime }$ positions of the beam electrons, and the white solid line is the analytical solution for the single electron obtained from Eq. (24). (a) The case of a matched electron beam injected into the undulator with optimal conditions for minimizing the betatron amplitude: $(⟨x_{e,0}⟩,⟨y_{e,0}⟩)=(-a_u/({\gamma }_0\mathrm{\Omega }),0)$ and $(⟨u_{x,0}⟩,⟨u_{y,0}⟩)=(0,-a_u)$. The exponential term is neglected. (b) Undulator structure is the same as in case (a), but a matched electron beam is injected with $(⟨x_{e,0}⟩,⟨y_{e,0}⟩)=(0,0)$ and $(⟨u_{x,0}⟩,⟨u_{y,0}⟩)=(0,0)$. (c) The case of an unmatched beam with optimal injection for the undulator structure the same as in case (a). (d) The case of a matched electron beam with optimal injection into the undulator [same as in (a)], with the exponential term included in the expressions for the transverse field as given by Eq. (18).

Figure 4

Normalized on-axis single electron radiation spectrum (in arbitrary units) as a function of normalized photon energy $\kappa$ for the case of a plasma undulator with $N_u=30$. The blue curve corresponds to the case of optimal injection with the exponential term in Eq. (18) neglected. The black dashed line shows the analytical expression Eq. (33). The red curve corresponds to the case of on axis injection with no initial electron transverse momentum, with the exponential term in Eq. (18) neglected. The green curve is the on-axis radiation spectrum of the electron injected optimally into the plasma undulator with the exponential term in Eq. (18) included.

Figure 5

Photon energy-angular spectrum as a function of both normalized photon energy $\kappa$ (horizontal axis) and normalized angle ${\gamma }_0\theta$ (vertical axis) for four cases: (a) electron beam is matched and optimally injected into the plasma undulator [with the exponential term in Eq. (18) neglected]; (b) electron beam is matched, but injected on-axis into the same undulator; (c) electron beam is injected optimally, but unmatched to the same undulator; (d) electron beam is injected optimally and matched to the undulator, with exponential term in Eq. (18) included. Undulator parameters and beam parameters are the same as Figs. 3–3, with $N_u=30$.

Figure 6

Photon energy spectrum, obtained by integrating the energy-angular photon spectrum in angle from $\theta =0$ to $\theta ={\theta }_{\text{natural}}$, given by Eq. (41), as a function of normalized photon energy $\kappa$ for the following cases: (blue curve) optimally injected to minimize the betatron motion, matched beam, with $0.1\mu \mathrm{m}$ normalized emittance; (green curve) optimally injected, matched beam, with $0.025\mu \mathrm{m}$ emittance; (red curve) optimally injected, unmatched beam, with $0.1\mu \mathrm{m}$ normalized emittance; (cyan curve) on-axis injection, matched beam, with $0.1\mu \mathrm{m}$ normalized emittance. The black curve shows the case of an ideal beam with no transverse emittance and no energy spread. (Note that, for plotting purposes, the curve representing the ideal beam case was multiplied by a factor of 0.5.) The undulator parameters are the same as in Fig. 3, i.e., the exponential term in the focusing force is neglected.

Figure 7

Photon energy-angular spectrum in arbitrary units as a function of normalized photon energy $\kappa$ (longitudinal axis) and normalized angle ${\gamma }_0\theta$ (vertical axis) for the case of optimally injected electron beam. The colored lines show the corresponding contours of the radiation ellipticity (values indicated in colorbar) as functions of normalized photon energy $\kappa$ and angle ${\gamma }_0\theta$.

Figure 8

On-axis photon spectrum (red curve and left-vertical axis) in arbitrary units for an electron injected on axis (not optimal). Blue lines show the ellipticity of the on-axis radiation (value on right-vertical axis). (For illustrative reasons, ellipticity is not shown for the spectral regions with intensity less than 5% of the maximum intensity.)