Longitudinal electron bunch profile reconstruction by performing phase retrieval on coherent transition radiation spectra

The application of phase-retrieval algorithms to the reconstruction of the longitudinal bunch profile of an electron bunch from the spectrum of the coherent transition radiation (CTR) it produces is considered. The development of a new algorithm for this application, the Bubblewrap algorithm, is described. Tests with synthetic data show successful reconstruction of the longitudinal profile of single and double electron bunches, provided that the CTR spectrum is known over a sufficiently wide range. The application of the Bubblewrap algorithm to the reconstruction of laser-accelerated electron bunch profiles from experimental data is demonstrated. The results are shown to be consistent with estimates of the bunch length obtained by other methods.

Figure 1

Outline of the Gerchberg-Saxton phase-retrieval algorithm. For further details on each step see the text. Dashed boxes indicate the steps that form projections of a candidate function onto the support constraints, ${\pi }_{\mathrm{supp}}$, and the Fourier-space modulus constraints, ${\pi }_{\mathrm{mod}}$, respectively.

Figure 2

Performance of the hybrid input-output algorithm for different values of the parameter $\beta$, when reconstructing the spectrum of a Gaussian of $2\mu \mathrm{m}$ rms length on a fixed support of length $15\mu \mathrm{m}$. The rms length of the reconstructed profiles is ${\sigma }_{\mathrm{rec}}$, while $E_{1000}$ shows the reconstruction error after 1000 iterations, as defined by Eq. (4). The spectrum is truncated at wavelengths above $7\mu \mathrm{m}$, leading to consistent underestimates of ${\sigma }_z$. Each reconstruction run consists of 990 iterations of the HIO algorithm followed by ten iterations of the Gerchberg-Saxton algorithm. Ten reconstructions were carried out for each value of $\beta$.

Figure 3

Reconstruction of sample asymmetric bunch profiles. Initial profiles are shown as dashed gray lines in (a) and (c). The mean of the reconstructed profiles—from ten reconstructions with randomized initial data—are shown as blue lines. The light blue shading represents three-$\sigma$ deviation from the mean. Profile (a) is a combination of half Gaussians of rms lengths $0.5\mu \mathrm{m}$ (to the left of $z=0$) and $1\mu \mathrm{m}$ (to the right); for profile (c) the lengths are correspondingly $1\mu \mathrm{m}$ and $2\mu \mathrm{m}$. The spectral amplitudes corresponding to the original and reconstructed profiles are shown in (b) and (d). Spectral data is withheld from the phase-retrieval algorithm for $\omega <2.7\times {10}^{14}\mathrm{rad}{\mathrm{s}}^{-1}$, which is the only region where discrepancies between the original and reconstructed spectra occur.

Figure 4

Reconstruction of synthetic bunch profiles comprising two Gaussians, where the peak height of the smaller one is half that of the larger one. Initial profiles are shown as dashed gray lines in (a) and (c). The mean of the reconstructed profiles is shown as blue lines, and light blue shading represents three-$\sigma$ deviation from the mean. In (a) both Gaussians have rms length $1\mu \mathrm{m}$, and their centers are separated by $3\mu \mathrm{m}$; in (c) the separation is the same, but the smaller Gaussian has an rms length of $2\mu \mathrm{m}$. The spectral amplitudes corresponding to the original and reconstructed profiles are shown in (b) and (d).

Figure 5

Reconstruction of a synthetic bunch profile comprising two Gaussians. The initial profile is shown as dashed gray lines in (a). The mean of ten reconstructed profiles is shown as a blue line, and light blue shading represents three-$\sigma$ deviation from the mean. The rms lengths of the two Gaussians are $0.5\mu \mathrm{m}$ and $1.5\mu \mathrm{m}$ for the larger and smaller one, respectively, and the separation is $15\mu \mathrm{m}$. The spectral amplitudes corresponding to the original and reconstructed profiles are shown in (b).

Figure 6

Experimental setup for characterization of coherent transition radiation across the visible, near-infrared, and terahertz ranges of the electromagnetic spectrum. See text for details.

Figure 7

Contributions to the relative pyroelectric detector calibration due to (a) grating reflectivity and dispersion (black dots); and (b) pyroelectric detector response (red crosses). The calibration factors plotted represent the inverse of system/detector sensitivity. Each data point represents a single pyroelectric detector. The significant variation between individual detector responses illustrates the need for reliable calibration.

Figure 8

Reconstruction of longitudinal bunch profile from experimentally measured CTR spectrum: (i) image of Lanex screen at the exit of the electron spectrometer; (ii) ten reconstructed profiles, the most representative of which is colored red; (iii) measured spectrum (thick black line, black crosses) and reconstructed spectra (colored lines), including retrieved phase of most representative profile (gray dots). Data for Shot 1930: $P=83\mathrm{mbar}$ $n_e=4.0\times {10}^{18}{\mathrm{cm}}^{-3}$); $L_{\mathrm{cell}}=6\mathrm{mm}$; $E_{\mathrm{peak}}=507\pm 92\mathrm{MeV}$ (FWHM); $Q_{\mathrm{peak}}=8.7\mathrm{pC}$; $Q_{\mathrm{total}}=34.6\mathrm{pC}$.

Figure 9

Reconstruction of longitudinal bunch profile from experimentally measured CTR spectrum: (i) image of Lanex screen at the exit of the electron spectrometer; (ii) ten reconstructed profiles, the most representative of which is colored red; (iii) measured spectrum (thick black line, black crosses) and reconstructed spectra (colored lines), including retrieved phase of most representative profile (gray dots). Data for Shot 2168: $P=82\mathrm{mbar}$ ($n_e=4.0\times {10}^{18}{\mathrm{cm}}^{-3}$); $L_{\mathrm{cell}}=14\mathrm{mm}$; $E_{\mathrm{peak}}=599\pm 47\mathrm{MeV}$; $Q_{\mathrm{peak}}=3.8\mathrm{pC}$; $Q_{\mathrm{total}}=22.5\mathrm{pC}$.

Figure 10

Reconstruction of longitudinal bunch profile from experimentally measured CTR spectrum: (i) image of Lanex screen at the exit of the electron spectrometer; (ii) ten reconstructed profiles, the most representative of which is colored red; (iii) measured spectrum (thick black line, black crosses) and reconstructed spectra (colored lines), including retrieved phase of most representative profile (gray dots). Data for Shot 2147: $P=82\mathrm{mbar}$ $n_e=4.0\times {10}^{18}{\mathrm{cm}}^{-3}$); $L_{\mathrm{cell}}=13\mathrm{mm}$; $E_{\mathrm{peak}}=585\pm 48\mathrm{MeV}$; $Q_{\mathrm{peak}}=2.8\mathrm{pC}$; $Q_{\mathrm{total}}=21.0\mathrm{pC}$.