# Segmented Composite Optical Parametric Amplification
## 4 Summary and Conclusions
The three nonlinear coupled Equations (1) can be simplified in the case of sum-frequency generation (\(\omega_{1}+\omega_{2}=\omega_{3}\)) if the incoming pump wave at frequency \(\omega_{1}\) is much stronger than the signal wave at frequency \(\omega_{2}\) so that its amplitude remains constant during evolution (\(A_{1}=\text{const}\)). This leads to the simplified equations system,  
\[i\partial_{z}\mathbf{C}(z)=\mathbf{H}\mathbf{C}(z), \tag{10}\]  
where \(\mathbf{C}(z)=\left[C_{2}(z),C_{3}(z)\right]^{T}\), \(C_{2}(z)=A_{2}(z)e^{i\Delta kz/2}\), \(C_{3}(z)=A_{3}(z)e^{-i\Delta kz/2}e^{-i\phi}\), and \(\phi\) is the complex phase of the pump wave amplitude, i.e., \(A_{1}=|A_{1}|e^{i\phi}\). The above space-dependent matrix \(\mathbf{H}(z)\) is given by  
\[\mathbf{H}(z)=\frac{1}{2}\left[ \begin{array}{cc}-\Delta k&\Omega\\ \Omega&\Delta k\end{array} \right], \tag{11}\]  
where here \(\Omega=2\widetilde{\Omega}|A_{1}|\). As a further step, one can map the the \(z\)-dependence onto a time dependence, \(z=ct\). By doing so, Equation (10) becomes the conventional time-dependent Schrodinger equation for a two-state atom driven by an oscillatory field in the rotating-wave approximation [37; 38; 39; 40]. The two amplitudes \(C_{2}\) and \(C_{3}\) correspond to the probability amplitudes of the ground state and the excited state. The off-diagonal element \(\Omega\) in Equation (11) is well known as the Rabi frequency, whereas the component \(\Delta k\) correlates with the detuning [37; 38]. The solution of Equation (10) is particularly straightforward when \(\Delta k=0\), i.e., when exact phase matching occurs, which corresponds to the resonance case for a two-state atom. With the following initial conditions (at \(z=z_{i}\))  
\[C_{2}\left(z_{i}\right)=1,\quad C_{3}\left(z_{i}\right)=0, \tag{12}\]  
which mean that the field at the sum frequency \(\omega_{3}\) is not present at the input, the solution is  
\[C_{2}\left(z\right) =\cos\left(\tfrac{1}{2}S\right), \tag{13a}\] \[C_{3}\left(z\right) =-i\sin\left(\tfrac{1}{2}S\right), \tag{13b}\]  
where \(S=\int_{z_{i}}^{z}\Omega(z^{\prime})\,dz^{\prime}\). Therefore, if \(S=0\) or an even-integer multiple of \(2\pi\), no energy transfer from the signal to the wave at the sum frequency \(\omega_{3}\) (called idler here) occurs. In contrast, full energy transfer signal \(\rightarrow\) idler takes place when \(S=\pi\) or an odd-integer multiple of \(\pi\). As a result, the conversion efficiency in the case of exact phase matching is easily affected by variations in the crystal length, temperature, wavelength, and pump intensity. Two robust alternatives to the phase-matched case may be given if one use the techniques of adiabatic evolution or composite pulses from nuclear magnetic resonance. Adiabatic evolution in a dynamical system occurs when an external perturbation of the system varies very slowly compared to its internal dynamics, allowing the system the time to adapt to the external changes. Mathematically, it means that for the entire dynamical evolution, the system remains at one of the system's eigenmodes. This analogy was developed by [PERSON] et al. [32; 33; 41; 42], who, in a set of experiments, demonstrate a robust broad bandwidth conversion with high efficiency for sum frequency generation from the near-IR into the visible spectrum. Composite pulses are solutions to arbitrary optimization problems in a quantum system, driven by an external radiation field. The basic idea is to improve the performance of single-pulse excitation processes by applying multi-pulse (i.e., \"composite pulse\") processes. In quantum physics, nearly all composite pulses use the relative phases between the constituent pulses to yield a better performance of the composite excitation process compared to the single-pulse excitation [43; 44].