# Ultrafast Mid-IR Laser Pulses Generation via Chirp Manipulated Optical Parametric Amplification
## 3 Theory on Broadband Pumped OPA
### Gain Mechanism of Idler Pulse  
According to the above illustrations, the gain mechanism of the idler pulse is dependent on not only the instantaneous wavelengths of the input pulses, but also on the phase-matching condition during the propagation. Firstly, we investigate how the pump and seed chirps affect the initial idler bandwidth in the infinitely thin crystal. The seed pulse is stretched to a duration of 60 fs and pump pulses with different bandwidths are stretched to a same duration of 50 fs. The seed pulse with a TL duration of 13 fs is stretched with a negative GDD of \(-180\) fs\({}^{2}\). The pump pulses with TL durations of 10 fs, 20 fs, 30 fs, and 50 fs are stretched with GDDs of \(\pm 90\) fs\({}^{2}\), \(\pm 165\) fs\({}^{2}\), \(\pm 215\) fs\({}^{2}\), and 0 fs\({}^{2}\), respectively. The time-dependent frequencies of the initial pump, seed, and idler pulses are plotted in Figure 5. Figure 5 denote the frequencies when both the input pulses are negatively chirped, and Figure 5 denote the frequencies when the pump pulse is chirped positively the while seed pulse is chirped negatively. In order to stretch different pump pulses to the same duration, a larger pump chirp  
Figure 4: Phase matching efficiency with respect to the pump and seed wavelengths in a \(\theta=20^{\circ}\) BBO crystal, the blue line denotes the linear chirp-compensation fitting. is required for a pump pulse with broader bandwidth, which explains the slopes of different solid lines in Figure 5. As given above, the temporal chirp of idler \(C_{i}\) is equal to \(C_{p}-C_{s}\). Now that the seed chirp \(C_{s}\) is fixed at a negative value (dashed line in Figure 5), \(C_{i}\) becomes \(C_{p}+|C_{s}|\), which means that a larger positive \(C_{p}\) corresponds to a larger \(C_{i}\). By comparing the dotted lines in Figure 5a,b, it is easily seen that the TL = 10 fs pump pulse with a GDD of 90 fs\({}^{2}\) (black solid line in Figure 5b) has the largest temporal chirp among all the studied case, hence resulting in the largest idler chirp as the black dotted line in Figure 5b. The initial idler pulse is produced in a time window where the input pump and seed pulses are temporally overlapped. Since the pump and seed pulses have similar durations after being chirped in all cases, the durations of the initially generated idler are also similar. According to the equation \(\omega_{i}(t)=\omega_{i0}+C_{i}t\), when the idler duration is fixed, the major factor that decides the idler bandwidth becomes its temporal chirp. As a result, the broadest initial idler bandwidth is obtained with oppositely chirped pump and seed pulses. As the initial pulses further propagate in the crystal, the nonlinear interaction between the pulses begins, during which the phase-matching condition plays an important role in deciding the idler performances. The phase-matching efficiency with respect to the pump chirp and idler wavelength is shown in Figure 6, the seed pulse is still chirped with a negative GDD of \(-\)180 fs\({}^{2}\). The initial idler bandwidth is included in the figure as the white dotted lines for reference, the bandwidth is defined as the wavelength components inside the temporal window of \(-\)50 fs \(\leq\) t \(\leq\) 50 fs. Since the temporal durations of all initial idler pulses are the same, this definition is a reasonable standard for the idler bandwidth comparison. Pump pulses with TL durations of 10 fs, 20 fs, 30 fs, and 50 fs are marked by the black, blue, red, and purple lines, respectively. All pump pulses are stretched to \(\sim\)50 fs, and the line lengths stand for the idler bandwidths in each case. Regarding the phase-matching efficiency, the bottom half of Figure 6 where the pump and seed pulses are chirped with the same sign is preferred, in which high efficiency phase matching can be achieved for a bandwidth up to 300 nm. The maximum phase-matching bandwidth is realized with a pump GDD of \(-\)40 fs\({}^{2}\), which is equivalent to the aforementioned linear fitting to the chirp-compensation scheme.