# Ultrafast Mid-IR Laser Pulses Generation via Chirp Manipulated Optical Parametric Amplification
## 3 Theory on Broadband Pumped OPA
### Simulations on BBP-OPA  
The performance of broadband-pumped chirp-manipulated OPA (referred to as BBP-OPA in the following text for simplicity) is quantified by numerically simulating the process with coupled-wave equations. Various pump bandwidths and chirps are studied for a thorough comparison. Details on the numerical model can be found in [107; 109]. The output signal and idler pulses are characterized by the full width at 1/e\({}^{2}\) maximum (FW1/e\({}^{2}\)M) bandwidth and energy-bandwidth product (EBP). Pump pulses with the energy of 300 \(\upmu\)J and TL durations of 10 fs, 20 fs, 30 fs, and 50 fs are employed. For simplicity in the following illustrations, the TL = 50 fs pump is referred to as the narrowband pump (NBP, although its bandwidth is considerably broad), as opposed to the broadband pump (BBP) with TL durations less than 30 fs. Since the current bandwidth broadening techniques inevitably stretch the femtosecond pulses, the pump will be used as positively chirped pulses, with the designed durations ranging from 50 fs to 120 fs. The 1400-nm seed pulse is obtained from our previous experiment [94], with a FWHM bandwidth of \(>\)200 nm and the energy of \(\sim\)1 \(\upmu\)J. The seed pulse is chirped with a negative GDD of \(-\)180 fs2. Footnote 2: The time-domain of the pump is \(\sim\)10 fs, which is the time domain of the pump. The simulation results are compared in Figure 7. As seen from the triangles in the figure, the signal bandwidth barely changes with different pump bandwidths, and the signal EBP even decreases when a broader pump bandwidth is employed. Nevertheless, in accordance to the above analysis, the BBP is evidently beneficial to the idler bandwidth and efficiency when the input pump and seed  
Figure 6: Phase-matching efficiency at different idler wavelengths and pump GDDs. The white lines represent the idler bandwidth within the temporal range of \(-\)50 fs \(\leq\) t \(\leq\) 50 fs. The black, blue, red, and purple lines indicate the pump pulses when their TL durations are 10 fs, 20 fs, 30 fs, and 50 fs respectively. pulses are oppositely chirped. The larger pump bandwidth increases both the idler bandwidth and EBP in Figure 7.  
With a longer stretched pump duration, the gain narrowing is less affective to the process, resulting in broader output bandwidths. Meanwhile, the longer pulse duration indicates a lower pump peak intensity, thus realizing lower conversion efficiencies and EBPs. The highest EBP is achieved with a BBP pulse when its stretched to 50 fs, the idler spectrum with 858-nm bandwidth is obtained with a EBP of 6.9 GW, which is an impressive improvement compared to the 314-nm bandwidth and 4.0-GW EBP. The 72.6% higher EBP is mainly originated from the broader idler bandwidth and shorter TL duration. The signal and idler spectra obtained with pump pulses stretched to 50 fs are presented in Figure 8. The input seed spectrum is plotted in Figure 8a, and the idler spectrum from an unchirped BBP-OPA is plotted in Figure 8b as the dashed lines for comparison. As shown in Figure 7a, the output signal bandwidths are similar with different pump bandwidth, hence only one spectrum is plotted as an example. The identical stretched pump duration is selected to ensure similar conversion efficiency for all systems, so the system performance can be directly revealed by the idler bandwidth. A BBO crystal with 2.5-mm thickness is used in all cases. Figure 8: (**a**) Input (dashed line) and output (solid line) spectra of the signal. (**b**) Output spectra of idler pulses from systems with different pump bandwidths. Idler spectrum from an unchirped OPA (black dashed line) is plotted for comparison. Figure 7: (**a**) Bandwidth and (**b**) energy-bandwidth product (EBP) variations of signal (triangles) and idler (circles) pulses with respect to the duration of the stretched pump pulses. The black, blue, red, and yellow lines denote pump pulses with TL durations of 10 fs, 20 fs, 30 fs, and 50 fs, respectively.