Sub-Single-Cycle Optical Pulse Train with Constant Carrier Envelope Phase

We describe the synthesis of periodic waveforms consisting of a train of pulses that are 0.83 cycles long and have an electric field pulse width of 0.44 fs using 7 Raman sidebands generated by molecular modulation in ${\mathrm{H}}_2$. We verify by optical correlation that the carrier-envelope phase is constant in these waveforms when they are synthesized from commensurate sidebands. The estimated overall shift of the carrier-envelope phase is less than 0.18 cycles from the first to the last pulse of nearly ${10}^6$ pulses in the pulse train.

Figure 1

Four-wave mixing signal at 301 nm in Xe as a function of the adjustment of the phase of a single sideband in progressive steps at the wavelengths of (a) 481 nm, 301 nm being generated with the sidebands $q=1$, 2, 3; (b) 602 nm, after adding $q=0$ to case (a); (c) 802 nm, after adding $q=-1$ to (b); and (d) 1203 nm, with all 6 sidebands included in the generation. Points: experimental data, Lines: simulation.

Figure 2

Cross-correlation signal (points) at 301 nm as a function of the relative time delay between one pulse consisting of sidebands of wavelength and the corresponding pulse energy ($\mu \mathrm{J}$ in brackets) of 1203 nm (34), 802 nm (117), 602 nm (207) with respect to a second pulse made of 481 nm (160), 401 nm (23), and 344 nm (7), respectively. The predicted signal (line) is calculated with the pulse energies shown here in brackets and normalized to the experimental value at zero time delay.

Figure 3

Cross-correlation signal (points) produced by two subsets of incommensurate sidebands at 298 nm obtained and plotted as a function of relative pulse delay in a manner similar to that in Fig. 2. Here one pulse has sidebands with wavelength and corresponding pulse energy ($\mu \mathrm{J}$ in brackets) of 1155 nm (16), 780 nm (188), 589 nm (240), and the second pulse has 473 nm (197), 396 nm (34), and 340 nm (3.7), respectively. Solid line is from calculation. Zero time delay is set to the center of the symmetric signal waveform.

Figure 4

Cross-correlation signal (points) at 267 nm as a function of relative time delay between one pulse consisting of the sidebands of wavelengths and corresponding pulse energies ($\mu \mathrm{J}$ in brackets) of 1203 nm (26), 802 nm (148), 602 nm (144), with respect to another consisting of 481 nm (113), 401 nm (33), 344 nm (11), and 301 nm (2.3), respectively. The solid line is the calculated trace using pulse energies shown in brackets.