## 6 Nonequivalent Influence of Pulses' GVM on the THG Efficiency under Cascading SHG
### Compensation of GVM Action by Choosing Phase Mismatching and Shifting of FW Carrier Frequency  
We see in the previous paragraph that the GVM between FW and TH may be the main physical mechanism restricting the THG efficiency. Therefore, we analyse briefly another (together with duration increasing) possible way to minimise (or remove at all) an influence of this GVM. Below we show that it is possible to implement by choosing the incident pulse phase distribution. Let us note that we do not consider the incident chirped pulse because it is of interest for another study. Our current interest is the influence of the frequency shift on the energy conversion efficiency in a medium with a quadratic nonlinear response. Previously, we have shown in [67] how it is possible to increase the energy conversion efficiency of the SHG process in a medium with the combined nonlinear response by choosing the appropriate incident frequency shift and the phase mismatching between FW and SH. Here, we provide a similar derivation for the THG process under the cascading SHG. The incident complex amplitude distribution is chosen as follows:  
\[A_{1}(0,t)=e^{-(t-0.5L_{1})^{2}}e^{i\varphi(t)},\ A_{2}(0,t)=A_{3}(0,t)=0. \tag{27}\]  
We transform the equations in (1) by using the following substitution of the complex amplitudes:  
\[A_{j}(z,t)=\tilde{A}_{j}(z,t)e^{j\varphi(t)},\ j=1,2,3. \tag{28}\]  
In fact, one can consider a more general case in which the frequency shifts differ from \(j\varphi(t)\), but this is outside of our current consideration. It is easy to write a new set of equations:  
\[ \begin{split}&\frac{\partial\tilde{A}_{1}}{\partial z}-2D_{1} \varphi^{\prime}(t)\frac{\partial\tilde{A}_{1}}{\partial t}+iD_{1}\frac{ \partial^{2}\tilde{A}_{2}}{\partial t^{2}}+i\gamma\Big{(}A_{1}^{*}\tilde{A}_{ 2}e^{-i\Delta_{21}kz}+\tilde{A}_{2}^{*}\tilde{A}_{3}e^{-i(\Delta_{21}k-\Delta_ {21}k)z}\Big{)}\\ &+i(-D_{1}(\varphi^{\prime}(t))^{2}+iD_{1}\varphi^{\prime\prime }(t))\tilde{A}_{1}=0,\\ &\frac{\partial\tilde{A}_{2}}{\partial z}+(v_{21}-4D_{2}\varphi^ {\prime}(t))\frac{\partial\tilde{A}_{2}}{\partial t}+iD_{2}\frac{\partial^{2} \tilde{A}_{2}}{\partial t^{2}}+i\gamma\Big{(}A_{1}^{2}e^{i\Delta_{21}kz}+2A_{ 1}^{*}\tilde{A}_{3}e^{-i(\Delta_{21}k-\Delta_{21}k)z}\Big{)}\\ &+i(-4D_{2}(\varphi^{\prime}(t))^{2}+2 iD_{2}\varphi^{\prime\prime }(t)+2v_{21}\varphi^{\prime}(t))\tilde{A}_{2}=0,\\ &\frac{\partial\tilde{A}_{3}}{\partial z}+(v_{31}-6D_{3}\varphi^ {\prime}(t))\frac{\partial\tilde{A}_{3}}{\partial t}+iD_{3}\frac{\partial^{2} \tilde{A}_{3}}{\partial t^{2}}+3i\gamma\tilde{A}_{1}\tilde{A}_{2}e^{i(\Delta_ {21}k-\Delta_{21}k)z}\\ &+i(-9D_{3}(\varphi^{\prime}(t))^{2}+3 iD_{3}\varphi^{\prime\prime }(t)+3v_{31}\varphi^{\prime}(t))\tilde{A}_{3}=0,\ 0<z\leq L_{z},0<t<L_{t}.\end{split}  \tag{29}\]  
Comparing the coefficients at the first derivative with respect to time in the first equation and the third one, we can choose the condition of synchronous motion of the pulses at FF and tripled frequency:  
\[\varphi(t)=\frac{v_{31}}{2(3D_{3}-D_{1})}t. \tag{30}\]  
Of course, inequality \(D_{1}\
eq 3D_{3}\) must be valid. Otherwise, it is impossible to compensate the GVM influence by choosing of frequency shift in the form (28). If the equality (30) is valid, then the set of equations in (29) takes the form  
\[