Generation of relativistic electron bunches with arbitrary current distribution via transverse-to-longitudinal phase space exchange

We propose a general method for tailoring the current distribution of relativistic electron bunches. The technique relies on a recently proposed method to exchange the longitudinal phase space emittance with one of the transverse emittances. The method consists of transversely shaping the bunch and then converting its transverse profile into a current profile via a transverse-to-longitudinal phase-space-exchange beam line. We show that it is possible to tailor the current profile to follow, in principle, any desired distributions. We demonstrate, via computer simulations, the application of the method to generate trains of microbunches with tunable spacing and linearly ramped current profiles. We also briefly explore potential applications of the technique.

Figure 1

Overview of the proposed technique to produce relativistic electron bunch with arbitrary current profile.

Figure 2

Rendition of a stratified longitudinal phase space generated downstream of the EEX beam line with an incoming transversely modulated beam.

Figure 3

Evolution of the transverse horizontal (blue), longitudinal (green), and vertical (red) emittances along the EEX beam line with parameters given in Table . The cyan symbols indicate the axial locations of the four dipoles (shown as rectangles and labeled $\mathrm{D}i$) and deflecting cavity (vertical line at $s\simeq 3\mathrm{m}$). The initial normalized emittance partition is $({\epsilon }_{x,0}^n,{\epsilon }_{y,0}^n,{\epsilon }_{z,0}^n)=(1,1,10)\mu \mathrm{m}$.

Figure 4

Fractional horizontal (left column) and longitudinal (right column) emittance dilutions at the exit of the exchanger beam line versus the incoming horizontal Courant-Snyder beam parameters. The rows correspond to different incoming normalized emittance partitions from top to bottom $({\epsilon }_{x,0}^n,{\epsilon }_{z,0}^n)=(10,1)$, (1,1), and (1,10) (all numbers are in micrometers). The longitudinal phase space parameters used in these calculations are a bunch length ${\sigma }_{z,0}=400\mu \mathrm{m}$, and a chirp $\mathcal{C}=6{\mathrm{m}}^{-1}$.

Figure 5

Example of conversion of a transverse modulation (a) into a modulation on the longitudinal phase space (d). The plots (a) and (c) show the initial horizontal and longitudinal phase space while the plots (b) and (d) show the corresponding final phase spaces downstream of the EEX beam line. The initial emittance partition is $({\epsilon }_{x,0}^n,{\epsilon }_{z,0}^n)=(1,10)\mu \mathrm{m}$.

Figure 6

Bunching factor (a), contrast ratio (b), and modulation wavelength (c) obtained for a longitudinally modulated beam produced via interception with thick mask upstream of the EEX beam line. The slit width is $w=100\mu \mathrm{m}$ and the slit separation $\delta w$ is variable. The parameters with subscript $m$ (and shown as blue traces) are computed using the longitudinal beam distribution downstream of the EEX beam line while the parameters with subscript $M$ (and shown as red traces) are obtained after the removal of the chirp associated to the beamlet (in this latter case, the beamlets are made upright). The parameters are calculated for various initial normalized emittance partitions $({\epsilon }_{x,0}^n,{\epsilon }_{z,0}^n)$ shown in the legend. The beam Lorentz factor for these simulations is $\gamma =30$.

Figure 7

Locus of C-S parameters that provide an upright longitudinal trace space for the beamlets (dashed solid line). The false-color map represents the emittance dilution $\delta {ϵ}_z$ as a function of initial C-S parameters and the solid white line indicates the limit where emittance dilution is 5%. The initial normalized emittance partition is $({\epsilon }_{x,0}^n,{\epsilon }_{z,0}^n)=(10,1)\mu \mathrm{m}$. The dashed white line is computed numerically and corresponds to Eq. (20).

Figure 8

Example of the generation of linearly ramped current profile from an initial uniform triangular distribution in the $(x,y)$ space (a) shown with corresponding final transverse distribution (b) and longitudinal phase space (c) downstream of the EEX beam line. The initial normalized emittance partition is $({\epsilon }_{x,0}^n,{\epsilon }_{z,0}^n)=(10,1)\mu \mathrm{m}$ and $({\alpha }_{x,0},{\beta }_{x,0},)=(2,12\mathrm{m})$. The current profiles downstream of the EEX beam line are shown in plots (d) and (e) for, respectively, $({\epsilon }_{x,0}^n,{\epsilon }_{z,0}^n)=(10,1)$ and $(1,10)\mu \mathrm{m}$. The incoming horizontal C-S parameters are a betatron function of 12 m and the $\alpha$ function was varied by an increment of 1 from $-4$ to $+2$ [corresponding to the seven traces shown in plots (d) and (e)]. The initial beam Lorentz factor for these simulations is $\gamma =30$. The head of the bunch corresponds to $t<0$.

Figure 9

Example of mask shape needed to generate a linearly ramped horizontal profile by interception of beam with transverse radial Gaussian density (a). The mask profile used is described by Eq. (22) and only macroparticles with $|y|\le \nu (x)$ are kept resulting in the transverse charge density shown in (b). The initial Gaussian horizontal profile (c) is transformed into a linear-ramped horizontal profile (d). For this example the initial transverse beam sizes are ${\sigma }_x=1.4\mathrm{mm}$ and ${\sigma }_y=2.5\mathrm{mm}$ and the parameters in Eq. (22) are $g_0=100$ and $g_1=3\times {10}^4$.

Figure 10

Overview of the photoinjector configuration used for realistic simulations of bunch shaping. The legend is “$\mathrm{D}i$” dipole magnets, “$\mathrm{Q}i$” quadrupole magnets, “$\mathrm{L}i$” solenoidal lenses. The geometry of the EEX beam line is detailed in Table .

Figure 11

Example of generation of modulated longitudinal trace space downstream of the EEX beam line. The four plots correspond to the four cases of initial parameters $(\delta w,w,\mathcal{F})$ of (100, 300, 0.92) [plot (a)], (100, 300, 1.25) [plot (b)], (50, 150, 0.75) [plot (c)], and (25, 75, 0.75) [plot (d)], where $\delta w$ and $w$ are, respectively, the slit edge-to-edge separation and width in micrometers, and $\mathcal{F}$ is a scaling factor for the quadrupoles Q1, Q2, Q3, and Q4 (when $\mathcal{F}=1$ the quadrupoles are tuned to match the horizontal C-S parameter to $({\alpha }_{x,0},{\beta }_{x,0})=(2,12\mathrm{m})$ upstream of the exchanger).

Figure 12

Bunch form factor associated to train of microbunches obtained downstream of the EEX beam line. The four traces correspond to four configurations indicated in the legend as “$\delta w-w-\mathcal{F}$” where $\delta w$ and $w$ are, respectively, the slit edge-to-edge separation and width in micrometers, and $\mathcal{F}$ is a scaling factor for the quadrupoles Q1, Q2, Q3, and Q4 [when $\mathcal{F}=1$ the quadrupoles are tuned to match the horizontal C-S parameter to $({\alpha }_{x,0},{\beta }_{x,0})=(2,12\mathrm{m})$ upstream of the exchanger].

Figure 13

Initial transverse densities on the photocathode surface (top row) and corresponding transverse densities (middle row) and horizontal profiles (bottom row) at $z=2.79\mathrm{m}$. The three columns, respectively, correspond to the case of a uniform-triangle distribution (left column), a circular distribution with a dipole moment (middle column), and a triangular-shaped distribution with a dipole moment (right column).

Figure 14

Example of longitudinal phase space (left) and associated linearly ramped current profile (b) obtained downstream of the EEX beam line. The beam Lorentz factor is $\gamma \simeq 33.4$ and the rms bunch length is $490\mu \mathrm{m}$. The initial distribution used for this calculation is the triangular-shaped distribution with a dipole moment (right column in Fig. 13).

Figure 15

Axial electric field (blue traces) behind a 1-nC bunch with Gaussian (a) and linearly ramped (b) current profiles (the bunch profiles are shown as red traces). In our convention $E_z<0$ corresponds to a decelerating field. For both cases the bunch is centered at $\zeta =0$ and the head corresponds to $\zeta >0$. The transformer ratio quoted in the text is computed as the ratio of $E_z(\zeta \simeq -1.3\mathrm{mm})$ over $|E_z(\zeta \simeq 0\mathrm{mm})|$.

Figure 16

Geometry of the ${\mathrm{TM}}_{110}$-like deflecting mode cavity used for the simulation presented in this paper. The superimposed false-color pattern represents the value of the $E_z$ field (the blue and red colors, respectively, correspond to the maximum and minimum values of the field).

Figure 17

Comparison of the analytical (solid lines) and numerical (squares) calculations of the $\kappa$-dependent elements of transfer matrix associated to the deflecting cavity considered in this paper. The analytical results are computed using Eq. (a2) while the numerical results are derived from tracking simulations with impact-t.