**3. Controlling the deposition of pump energy**

In this section, the realizability of a second novel technique that allows the simultaneous increase in the spectral bandwidth and optical conversion efficiency of OPCPA systems is discussed. This approach is based on a patent application by Deng and Krausz [36].

In the conventional OPCPA systems, similar to the one described in Section 2, the seed-pulse duration is designed to be a fraction of pump-pulse duration in order to maximize the energy conversion. Here as the seeds are strongly chirped, the temporal intensity profile of the pump pulses has to be nearly constant to ensure uniform amplification for the entire seed spectrum. Therefore, for pump pulses with Gaussian temporal profile, the seed pulses have to be considerably shorter than the pump pulses. Consequently, the pump energy is not consumed efficiently and the relative seed-to-pump pulse duration ratio will be a compromise between the amplification bandwidth and the conversion efficiency.

These deficiencies can be overcome if OPCPA seed pulses are linearly stretched to several times longer than the pump pulses. Subsequently, different fraction of seed pulses can be temporally overlapped with pump pulses and are amplified in different OPCPA stages.

This technique enables the controlled deposition of pump energy in the subsequent temporal/ spectral locations along the chirped seed pulse [36]. Furthermore, by controlling the amplifi‐ cation gain in each stage, the spatiotemporal profile of the pump pulses can be shaped into a flat-top pulse. By tuning the phase-matching angle of the crystal to the central wavelength of the seed pulse, the ultimate amplified spectrum can be shaped and a broader amplified bandwidth is gained. In addition, by reusing the pump energy after each amplification stage, the total conversion efficiency is increased.

## **3.1. Theoretical analysis**

conversion efficiency of >32% was achieved (**Figure 3(b)** and **(c)**), which to the best of our knowledge is the highest reported conversion efficiency for few-cycle OPCPA systems [9, 10, 29, 30]. No measurable superfluorescence background was observed when blocking the signal

The simulated boost efficiency in our design study is in good agreement with the experimental results. Quantitative comparison shows, however, that higher conversion efficiencies were yielded for a shorter crystal in the simulation than in the experiment. We relate the deviation from the theoretical prediction to a slight ellipticity in our pump beam, caused by the com‐ pressor of the Yb:YAG amplifier, which limited the effective interaction area between pump

**Figure 4.** (a) Retrieved temporal intensity of the compressed pulses after 12 reflections in a double-angle chirped-mir‐ ror compressor measured by SH-FROG. The pulse is compressed to 9.5 fs and holds Fourier transform limit of 5.7 fs. (c) The calculated GD of retrieved spectral phase for the pumped (blue curve) and unpumped (green curve) OPCPA

The 350 nm broad amplified signal measured with the Si-based spectrometer supports a transform-limited pulse duration of 5.7 fs. Preliminary compression, by using 12 reflections on double-angle chirped mirrors with −30 fs<sup>2</sup> GDD per reflection, resulted in a pulse duration of 9.5 fs (FWHM). The compressor had a total throughput of 80%. The retrieved temporal intensity profile and retrieved residual group delay (GD) of the pulses are shown in **Fig‐ ure 4(a)** and **(b)**. Our analytical study shows that the pulse can be compressed to 7 fs by adding the GD of a 0.5 mm thick fused silica to the measured GD of the pulse. However, to investigate the origin of the fine oscillation in the retrieved GD, a frequency resolved optical gating (FROG) measurement of the whole OPCPA chain was performed, but this time without any pumping. The comparison between two cases in **Figure 4(b)** shows that oscillations were enhanced by amplification but did not originate from the OPCPA phase [31]. The peak of the GD is at 760 nm, which coincides with the wavelength of Ti:Sa amplifier's pulses and the peak in the spectral intensity of the seed pulses after the HCF. Therefore, it can be concluded that the measured residual higher-order chirp is due to the self-phase modulation in the HCF, OPCPA phase, and the residual oscillations in group delay dispersion of the double-angle chirped-mirror compressor [32, 33]. The higher-order dispersion and the satellite pulses can be compensated

beam in front of the first stage.

62 High Energy and Short Pulse Lasers

and signal beams.

chains.

**Figure 5** shows simulation results for three different OPCPA designs. The simulation's input parameters are similar to the ones presented in Section 2.

In the first design, the residual pump energy after the first amplification stage is used to pump the second stage and ultimately the residual pump energy after the second stage is used to pump the third stage.

Due to the Gaussian shape of the pump in time and space, the energy extraction takes place primarily in the middle of the pump. Therefore, the wings are mostly left unaffected with a signature of energy back conversion at the center, due to the fact that this part of the pump possesses the highest peak intensity. In this design, the OPCPA pump-to-signal conversion efficiency is increased to 43% compared to the OPCPA system demonstrated in Section 2 (**Figure 5(b)** and **(c)**).

**Figure 5.** (a) Three designs are discussed in the main text. Design 1 consists of three OPCPA stages, where the residual pump energy after each amplification stage is reused in a subsequent OPCPA stage. Design 2 consists of two OPCPA stages. Here, the seed pulses are temporally stretched to twice the pump-pulse duration, and the residual pump ener‐ gy after the first OPCPA stage is reused in the second stage. In Design 3, the seed pulses are temporally stretched to triple of the pump-pulse duration, and the residual of the pump energy is reused after each amplification stage. (b) Calculated amplified signal energy over the crystal length and their corresponding spectra (c) for the three different designs [7].

In the second design, the seed pulses are stretched temporally to twice the pump-pulse duration. The blue part of the seed spectrum is amplified in the first OPCPA stage by adjusting the temporal overlap between pump and seed pulses. Subsequently, the pump pulses after the first stage are reused to amplify the red part of the spectrum at the second stage. In this design, the pump-to-signal conversion efficiency reaches 45% indicating the good pump-energy extraction while the beam quality of the amplified signal is maintained (**Figure 5(b)** and **(c)**).

The 45% conversion efficiency achievable from the second design is not drastically different from the 38.6% efficiency achievable from the case where the seed and pump pulses have the same pulse duration (as discussed in Section 2) as the pump pulses after the first amplification stage in both cases maintain a good spatiotemporal profile.

The gain and the shape of the amplified spectrum can be further optimized by adjusting the phase-matching angles of the crystal in each stage to tune the amplification for the selected part of the spectrum, which is not investigated in this study.

In the third design, the seed pulses are stretched three times the pump pulses and amplified in three subsequent OPCPA stages, while the residual pump energy of the preceding stage is used to pump the subsequent stage. The bluest frequencies of the seed spectrum are amplified in the first stage, while the reddest frequency components are amplified in the third OPCPA stage. The pump-to-signal conversion efficiency in this design reaches 57%, which is a noticeably higher value compared to the other designs (**Figure 5(b)** and **(c)**). The spectral narrowing for Designs 2 and 3 is caused by a suboptimal stretching factor of the input signal and phase-matching angle of the crystal. The optimizations of these parameters are cumber‐ some in simulation but straightforward in an experimental setup.

#### **3.2. Experimental setup**

In the first design, the residual pump energy after the first amplification stage is used to pump the second stage and ultimately the residual pump energy after the second stage is used to

Due to the Gaussian shape of the pump in time and space, the energy extraction takes place primarily in the middle of the pump. Therefore, the wings are mostly left unaffected with a signature of energy back conversion at the center, due to the fact that this part of the pump possesses the highest peak intensity. In this design, the OPCPA pump-to-signal conversion efficiency is increased to 43% compared to the OPCPA system demonstrated in Section 2

**Figure 5.** (a) Three designs are discussed in the main text. Design 1 consists of three OPCPA stages, where the residual pump energy after each amplification stage is reused in a subsequent OPCPA stage. Design 2 consists of two OPCPA stages. Here, the seed pulses are temporally stretched to twice the pump-pulse duration, and the residual pump ener‐ gy after the first OPCPA stage is reused in the second stage. In Design 3, the seed pulses are temporally stretched to triple of the pump-pulse duration, and the residual of the pump energy is reused after each amplification stage. (b) Calculated amplified signal energy over the crystal length and their corresponding spectra (c) for the three different

In the second design, the seed pulses are stretched temporally to twice the pump-pulse duration. The blue part of the seed spectrum is amplified in the first OPCPA stage by adjusting the temporal overlap between pump and seed pulses. Subsequently, the pump pulses after the first stage are reused to amplify the red part of the spectrum at the second stage. In this design, the pump-to-signal conversion efficiency reaches 45% indicating the good pump-energy extraction while the beam quality of the amplified signal is maintained (**Figure 5(b)** and **(c)**).

The 45% conversion efficiency achievable from the second design is not drastically different from the 38.6% efficiency achievable from the case where the seed and pump pulses have the

pump the third stage.

64 High Energy and Short Pulse Lasers

(**Figure 5(b)** and **(c)**).

designs [7].

The seed pulses of the OPCPA system described in Section 2 are stretched after the first amplification stage by using an 8 mm thick SF57 plate at Brewster's angle. The blue frequencies of the seed spectrum were amplified to 4 W by adjusting the temporal delay between the seed and pump pulses and adjusting the phase-matching angles of the BBO crystal (blue curve in **Figure 6(a)**). In the next OPCPA stage, the amplification is moved to the second half of the seed spectrum gaining 6 W of the total amplification (**Figure 6(a)**, pink curve).

**Figure 6.** (a) The experimental demonstration of Design 2. The seed pulses are heavily stretched by using an 8 mm thick SF57 plate. The higher frequencies in the spectrum are amplified first (blue curve). The residual of the pump en‐ ergy is used to amplify the lower frequency components of the spectrum (pink curve). (b) The gray curve shows the amplified spectrum in the similar OPCPA system without any spectral stacking (gray curve) compared to the ampli‐ fied spectrum shown in (a) (blue curve). (c) Further stretching of the seed pulses by using a 12 mm thick SF57 plate results in appearing of a hole in the amplified spectrum (purple curve) [7].

The same setup, after removing the bulk stretcher, resulted in 6.2 W of amplification after the third stage. As shown in **Figure 6(b)**, the amplified spectrum of the stacked OPCPA is more uniform due to the even amplification gain. The red wing of the spectrum carries more energy in the stacked OPCPA compared to the amplified spectrum achieved from the system discussed in Section 2. Moreover, the spectral spike at 780 nm is heavily suppressed. **Fig‐ ure 6(c)** shows amplified spectrum for a similar system but with a larger stretching factor. Here, after stretching the seed pulses using a 12 mm thick SF57 plate, 5 W of the average power was obtained. It can be seen that the amplified spectrum contains a hole, leaving the 8 mm thick SF57 plate, the optimum thickness for temporal stretching of the seed pulses.

Temporal stretching of the signal pulses to twice the pump-pulse duration, demonstrated in this section experimentally, did not show further increase in the OPCPA conversion efficiency compared to the scheme realized in Section 2. This similarity in the conversion efficiency is due to the fact that the spatiotemporal quality of the residual pump pulses after one amplifi‐ cation stage is preserved for both cases.

However, it is shown analytically that the further temporal stretching of seed pulses results in the increase in the conversion efficiency, as the spatiotemporal quality of the remaining pump pulses after two amplification stages is preserved just for the case of heavily chirped input seed.
