# Ultrafast Mid-IR Laser Pulses Generation via Chirp Manipulated Optical Parametric Amplification
## 1 Introduction  
In the past decades, femtosecond laser pulses have become effective tools in various fields including high-order harmonic generation (HHG) [1; 2; 3; 4; 5; 6], strong-field ionization [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17], and time-resolved spectroscopy [18; 19; 20; 21; 22; 23; 24] due to their ultrashort pulse duration and high peak intensity. Benefiting from the Kerr lens mode-locking (KLM) [25; 26] and chirped pulse amplification (CPA) [27; 28; 29], the Ti:sapphire lasers have developed rapidly towards shorter pulse durations and higher pulse energies. However, restrained by the bandwidth of the gain medium, femtosecond pulses from solid-state laser systems are mostly centered at wavelengths shorter than 1 \(\upmu\)m, which cannot meet the requirements of potential applications. Particularly, HHG is a well-accepted method of producing fully coherent light in the extreme ultraviolet (XUV) to X-ray regions, which utilizes femtosecond laser pulses as the driving source. During the HHG process, the maximum obtainable photon energy follows the cutoff law \(\rm E_{c}=I_{P}+3.17\times U_{P}\), where \(\rm I_{P}\) is the ionization potential and \(\rm U_{P}\) is the ponderomotive energy [1; 30; 31]. Since \(\rm U_{P}\) scales with the square of the laser wavelength \(\lambda^{2}\), using a driving pulse with a longer center wavelength is a practical approach to the generation of harmonics with higher photon energy. Besides, the longer wavelength laser source also meets the strict requirement of tunneling ionization with the Keldysh parameter \(\gamma\ll 1\), where the electron is ionized from the barrier formed by the combination of the Coulomb potential and quasi-static laser field. Moreover, using tunneling as fundamental concept,a lot of strong field phenomena and applications such as photoelectron holography [32], electron correlation in double ionization [33; 34] and molecular orbital tomography [35; 36; 37; 38] can be interpreted or realized. Motivated by the applications, efforts have been made to exploit ultrashort driving fields with tunable center wavelengths. Owing to the capability of producing visible (VIS) to mid-infrared (Mid-IR) pulses with high single-pass gain and ultrabroad bandwidth, optical parametric amplification (OPA) has been a promising source of tunable laser pulses in ultrafast research [39; 40]. OPA is a parametric process based on the second-order nonlinearity of a crystal, during which the energy of a high-frequency pump pulse transfers to a low-frequency seed pulse through difference frequency generation (DFG). The seed pulse is amplified to the signal and the pump pulse gets depleted, meanwhile an idler pulse with the frequency of \(\omega_{i}=\omega_{p}-\omega_{s}\) is generated. Energy conservation \(\hbar\omega_{p}=\hbar\omega_{s}+\hbar\omega_{i}\) and momentum conservation (also known as phase-matching) \(\overset{\rightarrow}{\hbar k}_{p}=\hbar\overset{\rightarrow}{k}_{s}+ \hbar\overset{\rightarrow}{k}_{i}\) have to be fulfilled in the three-wave interactions. Comprehensive overviews on the OPA principle and its development can be found in [41; 42; 43; 44; 45; 46]. In an ultrafast optical parametric amplifier, a critical limit on the gain bandwidth is the phase-mismatch (or wave-vector mismatch) \(\Delta k=\overset{\rightarrow}{k}_{p}-\overset{\rightarrow}{k}_{s}- \overset{\rightarrow}{k}_{i}\). In order to compress the obtained pulses to ultrashort or even few-cycle durations, the phase-mismatch has to be small values for a broad range of signal/idler wavelengths. Usually, the crystal angle \(\theta\) is selected to satisfy phase-matching for the center wavelengths, i.e., \(\Delta k_{0}=\overset{\rightarrow}{k}_{p0}-\overset{\rightarrow}{k}_{s0}- \overset{\rightarrow}{k}_{i0}=0\), hence the phase-mismatch is calculated to the first order.