Phase Diversity Electro-optic Sampling: A new approach to single-shot terahertz waveform recording

Abstract

Recording electric field evolution in single-shot with THz bandwidth is needed in science including spectroscopy, plasmas, biology, chemistry, Free-Electron Lasers, accelerators, and material inspection. However, the potential application range depends on the possibility to achieve sub-picosecond resolution over a long time window, which is a largely open problem for single-shot techniques. To solve this problem, we present a new conceptual approach for the so-called spectral decoding technique, where a chirped laser pulse interacts with a THz signal in a Pockels crystal, and is analyzed using a grating optical spectrum analyzer. By borrowing mathematical concepts from photonic time stretch theory and radio-frequency communication, we deduce a novel dual-output electro-optic sampling system, for which the input THz signal can be numerically retrieved-with unprecedented resolution-using the so-called phase diversity technique. We show numerically and experimentally that this approach enables the recording of THz waveforms in single-shot over much longer durations and/or higher bandwidth than previous spectral decoding techniques. We present and test the proposed DEOS (Diversity Electro-Optic Sampling) design for recording 1.5 THz bandwidth THz pulses, over 20 ps duration, in single-shot. Then we demonstrate the potential of DEOS in accelerator physics by recording, in two successive shots, the shape of 200 fs RMS relativistic electron bunches at European X-FEL, over 10 ps recording windows. The designs presented here can be used directly for accelerator diagnostics, characterization of THz sources, and single-shot Time-Domain Spectroscopy.

Fig. 1. Principle and limitations of classical single-shot THz waveform recorders using time-to-spectrum conversion (also known as spectral decoding).

a Principle. The electric field pulse shape modulates the birefringence of a Pockels electro-optic crystal. A probe chirped laser pulse (with stretched duration τ_w) is then intensity-modulated in single shot after passing the Pockels crystal, the quarter and half-wave plates (QWP and HWP) and the polarizing beam-splitter (PBS). Because of the laser chirp, the input temporal shape is expected to be “replicated” in the laser spectrum recorded by the grating OSA. b, c Fundamental time-resolution limitation of the method (numerical simulation assuming a perfect crystal with infinite bandwidth). The method is unreliable (i.e., strong deformations occur) when the input THz pulse is shorter than τ_R = 0.7 ps, although a 39 fs femtosecond laser is used. d Resolution limitation of the classical method (orange, from Eq. 1). An objective of DEOS is to remove this limitation, and obtain a resolution that does no more degrades when the duration of the analysis window τ_w is increased. The probe laser compressed duration τ_L (black line) is given for reference. Laser parameters: 1030 nm wavelength and 40 nm FWHM bandwidth (i.e., τ_L = 39 fs compressed laser pulse duration). τ_w = 10 ps FWHM. See Table 1 for crystal orientations, and “Materials and methods” for details).

Fig. 2. Principle of Phase Diversity Electro-optic Sampling, DEOS, for single-shot recording of THz electric fields.

a Experimental design: The input THz field evolution modulates a chirped laser pulse, and the DEOS design provides two outputs that contain different information. Using this two-output design (see Table 1 for details), the recorded information is sufficient for removing the problem, i.e., retrieving the input signal with high resolution. b−d Main steps of our reconstruction method (numerical simulation). b Raw electro-optic sampling (EO) signals (optical spectra after background subtraction and normalization). c Transfer functions H₁ and H₂ corresponding to the two polarization outputs showing the phase diversity operation. d Input signal retrieved from the recorded OSA signals [displayed in (b)] and the transfer functions H_1,2(Ω) [displayed in (c)], using the MRC algorithm (Eq. 10). Laser parameters close to those of the European XFEL experiment presented hereafter (1040 nm wavelength and 40 nm FWHM bandwidth, τ_w = 5 ps FWHM).

Fig. 3. Single-shot recording of free-propagating terahertz pulses over a window of the order of 20 ps.

a DEOS experimental setup. ZnTe: 1 mm-thick, 110-cut Zinc Telluride crystal, HWP: Half-wave plates, QWP: Quarter-wave plate, PBS: Wollaston polarizing cube beam-splitter. The beams emerging from the PBS are in the plane perpendicular to the figure. b Raw camera image containing the single-shot spectra of the two polarization outputs S_1,2n, and the unmodulated laser spectrum S_0n. c, d EO signals on two polarizations channels (after background subtraction and normalization, see “Materials and methods”. e Single-shot input signal retrieved from (c) and (d) using the DEOS phase-diversity-based algorithm (red). Green trace: actual input, obtained using scanned electro-optic sampling. Inset: Fourier spectra of the two terahertz signals. Note that the classical (i.e., single channel) spectral decoding method would just provide the deformed signals (c) or (d), depending on the channel used. More generally, a classical single-channel method would provide an input signal with good fidelity only if its bandwidth is small compared to the location of the first transfer function zero (dashed lines in the inset of (e)), or if the signal bandwidth is within one of the lobes in the frequency response in Fig. 2b.

Fig. 4. Fit providing the reconstruction parameter B from a single-shot recording.

Dots: Fourier spectra $\mid {\tilde{Y}}_{1,2}\left(\mathrm{\Omega }\right)\mid$ of experimental data before reconstruction. Lines: spectra $\mid {\tilde{Y}}_{1,2}^{\mathrm{retr}}\left(\mathrm{\Omega }\right)\mid$ computed from the retrieved input. ${\tilde{Y}}_{1,2}^{\mathrm{retr}}\left(\mathrm{\Omega }\right)$ are fitted on ${\tilde{Y}}_{1,2}\left(\mathrm{\Omega }\right)$ using B as a free parameter. Note the presence of interleaved zeros, which play a key role in the retrieval approach. Same data and color codes as for Fig. 3c, d.

Fig. 5. Electron bunch shapes recorded at the European X-ray Free-Electron Laser (EuXFEL).

a Picture from inside the 3 km-long accelerator tunnel. Our DEOS setup (see Fig. 6) is placed just upstream of the picture, after the first bunch compressor. b Timing of the electron bunches in the conditions of the experiment. c Electro-optic signals Y_1,2 of a single bunch before reconstruction. d Reconstructed electric field. Shaded areas: superposition of single-shot curves, color curves: average over 255 bursts. e Electro-optic signal of two bursts (i.e., 800 electron bunches in total). f Shape of one bunch (with bunch number 200 within the burst) versus burst number. g Arrival time versus bunch number. Shaded areas: RMS arrival time fluctuations, color curve: average over 255 bursts. The EO data are low-pass filtered to 2.5 THz.

Fig. 6. DEOS measurement of the Coulomb field created by relativistic electron bunches at the European X-ray Free-Electron Laser (Eu-XFEL).

A laser beam probes the electric field at a distance D = 5 mm from the electron beam. HWP: half wave plate, QWP, quarter wave plate, PBS, fiber-based polarizing beam-splitter. Blue lines indicate polarization-maintaining fibers (PM980) and green lines indicate single-mode fibers (HI1060). The probe laser is reflected on a Gallium Phosphide (GaP) crystal back side. The spectrum readout is performed using a KALYPSO linear array camera operating at 1.13 MHz line rate^,. Details of laser transport and KALYPSO focusing optics are not shown—see ref.  for further details