Quasi-half-cycle pulses of light from a tapered undulator

Strong-field few-cycle terahertz (THz) pulses are an invaluable tool for engineering highly nonequilibrium states of matter. A scheme is proposed to generate quasi-half-cycle GV/m-scale THz pulses with a multikilohertz repetition rate. It makes use of coherent spontaneous emission from a prebunched electron beam traversing an optimally tapered undulator. The scheme is the further development of the slippage control in free-electron lasers [T. Tanaka, Phys. Rev. Lett. 114, 044801 (2015)]. An explicit condition for the spectral amplitude of undulator radiation and a phase condition for the electron density distribution, required for the generation of desired pulses, are presented. The amplitude condition is met by proper undulator tapering, and a generic optimal undulator profile is found analytically. In order to meet the phase condition, the distance between the adjacent bunches is varied according to the instantaneous resonant undulator wavelength. A 3D analytical theory is complemented by a detailed numerical study based on a direct solution to the 3D wave equation.

Figure 1

Schematic illustration of the Fourier decompositions (a) and (c) of the quasi-half-cycle (b) and frequency-chirped (d) pulses, respectively. The quasi-half-cycle pulse is formed by the superposition of in-phase sinusoidal components, whereas the frequency-chirped pulse is composed of the sinusoidals with a nonlinear phase shift in between them. The amplitudes of the Fourier components are not to scale.

Figure 2

Schematic of a tapered undulator and a prebunched beam. The distance between the bunches $s$ changes as the radiation wavelength along the undulator impulse $\lambda$.

Figure 3

Typical variation of $g(z)$ and $k(z){\sigma }_b^2/2R(z)$ along the undulator for two trial tapers $f_u=z/L_u$ and $f_u=(1-z/L_u)$ referred to as the positive and negative ones. The bunch and undulator parameters are discussed in detail in Sec. 5.

Figure 4

Comparison of the spectra of the single-cycle and 1.5-cycle pulses to the on-axis undulator spectrum. The undulator profile is depicted in the inset. The undulator spectrum is calculated numerically using (11) but neglecting diffraction. The wavelength for each spectrum is normalized in the way that the peaks of the spectra are aligned. The wavelengths corresponding to the peak value of the undulator spectrum ${\lambda }_{\mathrm{peak}}^{\mathrm{und}}$ and half of it ${\lambda }_{1/2}^{\mathrm{und}}$ are presented as well.

Figure 5

Optimal undulator field envelope and radiation frequency vs distance in the undulator. $L_u=70\mathrm{cm}$.

Figure 6

Distances between the adjacent bunches and the resonant wavelength (17) along the undulator. The corresponding temporal separation between the bunches is also presented.

Figure 7

Top left: Density plot of $E_{\omega }$ generated by a single bunch as a function of the frequency and normalized radial distance $r/{\sigma }_b$; top right: density plot of $E(t)$ generated by a single bunch as a function of the time and $r/{\sigma }_b$; bottom left: $E_{\omega }$ generated by a prebunched beam as a function of the frequency and $r/{\sigma }_b$; bottom right: $E(t)$ generated by a prebunched beam as a function of the time and $r/{\sigma }_b$.

Figure 8

Top left and top right: On-axis $E_{\omega }$ and $E(t)$ generated by a single bunch vs the frequency and time, respectively; bottom left and bottom right: on-axis $E_{\omega }$ and $E(t)$ generated by a prebunched beam vs the frequency and time, respectively.

Figure 9

Spectral and temporal distributions of the undulator emission for the reversed (deliberately wrong) orientation of optimal tapering, $f_u^{\mathrm{opt}}(L_u-z)$.

Figure 10

On-axis $E(t)$ vs the time taking into account the spread of the arrival time of the bunches. The results for ten shots and the averaged value are depicted.

Figure 11

Generic undulator profile vs longitudinal distance $z$ is schematically depicted in the upper plot. The bottom plot shows corresponding resonant wavelengths (a4) and resonances around them. For $\xi <{\xi }_A$ and $\xi >{\xi }_E$, the solution to (a1) is given by (a2). For wavelengths $\xi$, whose resonances are completely within the integration region and isolated such as the stationary points $C$ and $C^{\prime }$, the solution (a3) is applicable. For the end points $A$ and $B$, the value of the integral is half of (a3). In the vicinity of the degenerated point $E$, ${\psi }^{\prime \prime }(z_E)=0$, Eq. (a10) should be used.

Figure 12

Illustration of the ponderomotive phase behavior for a parabolic undulator profile $f_u=1-(2z/L_u-1{)}^2$.

Figure 13

Comparison of the analytical solutions (a2), (a3), and (a10) to the numerical integration of Eq. (a1).