## 2 Overview on the Optical Parametric Amplification Process  
The basic working principles of parametric amplifiers are briefly described here. For a comprehensive overview, the reader is addressed to [4]. OPA is a well-known nonlinear optical process occurring in second-order non-linear crystals and involving energy transfer among beams at different frequencies [5; 6]. A simple sketch of the process is presented in Figure 1: energy is transferred from a high intensity beam (called the pump beam, at frequency \(\omega_{p}\)) to a lower intensity beam (called the signal beam, at frequency \(\omega_{s}\)), thus resulting in an amplification of the signal. During the pump-beam interaction, a third beam is generated (called the idler beam, at frequency \(\omega_{i}\)) in order to fulfill energy conservation. For a pump frequency higher than the signal and idler frequencies, the conversion rule is expressed as \(\hbar\omega_{p}=\hbar\omega_{s}+\hbar\omega_{i}\). In the case of OPA systems working in the IR, extremely powerful laser sources in the visible/near-IR spectral range can be used for pumping a parametric process of downconversion into longer wavelengths. Parametric amplification takes place all along the non-linear media, and it is strongly affected by propagation. Indeed, pump-to-signal/idler photon conversion occurs locally in the material with the local phase of the three fields being coupled by a precise relationship: \(\phi_{p}=\phi_{s}+\phi_{i}\). Then, the boosted signal and idler beams propagate inside the non-linear crystals. During propagation, they overlap with other photon contributions arising from different sites, resulting in a spatial interference. To observe a macroscopic effect of amplification, a constructive interference regime has to be achieved. Constructive interference can be achieved by properly tuning the phase velocities of the propagating waves. In particular, the following condition, known as the phase-matching condition, must be satisfied for the pump, signal and idler wavevectors (respectively, \(\vec{k_{p}}\), \(\vec{k_{s}}\), \(\vec{k_{i}}\)): \(\vec{k_{p}}=\vec{k_{s}}+\vec{k_{i}}\). In order to fulfill the phase-matching condition, birefringent non-linear media, with uniaxial or biaxial optical properties, are exploited. In this case, the phase velocity of each beam inside the crystal is defined according to the orientation of the polarization and to the direction of propagation. In general, beams with different polarizations and different propagation directions are used. Polarization and propagation direction are the two main degrees of freedom available to the users for  
Figure 1: (**a**) Power flow from the pump beam to the signal beam in optical parametric amplification; (**b**) the same optical parametric amplification process can be viewed as stimulated emission of signal photons from a virtual level excited by the pump. optimizing the OPA systems. In particular, by properly choosing these two parameters, phase-matching conditions can be simultaneously fulfilled for a very broad range of signal frequencies, thus resulting in a broadband amplification. In this sense, two interaction types can be distinguished depending on the polarization states of the three incoming waves: Type-I (signal and idler propagating with the same polarization, either ordinary or extraordinary) and Type-II (signal and idler propagating with orthogonal polarization, either ordinary or extraordinary). Type-I phase-matching provides signal amplification on a broad spectral region. However, this interaction type is typically associated with large spectral dispersion. On the other hand, Type-II phase-matching results in a narrower phase-matching band, suitable for delivering non-dispersed signal pulses. Concerning the propagation direction, the non-collinear pump-signal configuration was proven to ensure suitable phase-matching on an extended signal band, thus enabling a broadband parametric amplification. Broadband OPAs are ideal candidates for amplifying ultrashort pulses.