## 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
\begin{split}&\frac{\partial\tilde{A}_{1}}{\partial z}-\frac{D_{1}v_{31}}{3D _{3}-D_{1}}\frac{\partial\tilde{A}_{1}}{\partial t}+iD_{1}\frac{\partial^{2} \tilde{A}_{1}}{\partial t^{2}}+i\gamma\Big{(}\tilde{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\frac{D_{1}v_{31}^{2}}{4(3D_{3}-D_{1})^{2}}\tilde{A}_{1}=0,\\ &\frac{\partial\tilde{A}_{2}}{\partial z}+(v_{21}-\frac{3D_{2}v_{31 }}{3D_{3}-D_{1})}\frac{\partial\tilde{A}_{2}}{\partial t}+iD_{2}\frac{\partial ^{2}\tilde{A}_{2}}{\partial t^{2}}+i\gamma\Big{(}\tilde{A}_{1}^{*}e^{i\Delta_ {21}kz}+2\tilde{A}_{1}^{*}\tilde{A}_{3}e^{-i(\Delta_{21}k-\Delta_{21}k)z} \Big{)}\\ &-i\left(-D_{1}\frac{v_{31}^{2}}{(3D_{3}-D_{1})^{2}}+\frac{v_{31} \Delta_{3}}{3D_{3}-D_{1}}\right)\tilde{A}_{2}=0,\\ &\frac{\partial\tilde{A}_{3}}{\partial z}-\frac{D_{1}v_{31}}{3D_{3}- D_{1}}\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}\\ &+3i\frac{v_{31}^{2}(3D_{3}-D_{1})^{2}}{4(3D_{3}-D_{1})^{2}}\tilde{A} _{3}=0,\ 0<z\leq L_{z},0<t<L_{t},\end{split}  \tag{31}\]which means a presence of the additional phase mismatching between the interacting waves. Further, we make another substitution  
\[\bar{A}_{1} =\bar{A}_{1}e^{-iD_{1}\frac{v_{31}^{2}}{4(3D_{3}-D_{1})^{2}}z},\] \[\bar{A}_{2} =\bar{A}_{2}e^{i\Big{(}-D_{2}\frac{v_{31}^{2}}{(3D_{3}-D_{1})^{2}}+ \frac{v_{31}v_{31}}{3D_{3}-D_{1}}\Big{)}z},\] \[\bar{A}_{3} =\bar{A}_{3}e^{3i\frac{v_{31}^{2}(3D_{3}-2D_{1})}{4(3D_{3}-D_{1})^ {2}}z},\]  
and for new functions, we write the following set of equations:  
\[\frac{\partial\bar{A}_{1}}{\partial z}-\frac{2D_{1}v_{31}}{6D_{3}- 2D_{1}}\frac{\partial\bar{A}_{1}}{\partial t}+iD_{1}\frac{\partial^{2}\bar{A} _{1}}{\partial t^{2}}+i\gamma\Big{(}\bar{A}_{1}^{*}\bar{A}_{2}e^{-i\bar{A}_{2 }\kappa z}+\bar{A}_{2}^{*}\bar{A}_{3}e^{-i(\bar{A}_{31}k-\bar{A}_{21}k)z}\Big{)} =0,\] \[\frac{\partial\bar{A}_{2}}{\partial z}+(v_{21}-\frac{4D_{2}v_{31} }{6D_{3}-2D_{1}})\frac{\partial\bar{A}_{2}}{\partial t}+iD_{2}\frac{\partial^ {2}\bar{A}_{2}}{\partial t^{2}}+i\gamma\Big{(}\bar{A}_{1}^{2}e^{i\bar{A}_{2} \kappa z}+2\bar{A}_{1}^{*}\bar{A}_{3}e^{-i(\bar{A}_{31}k-\bar{A}_{21}k)z}\Big{)} =0, \tag{32}\] \[\frac{\partial\bar{A}_{3}}{\partial z}-\frac{2D_{1}v_{31}}{6D_{3} -2D_{1}}\frac{\partial\bar{A}_{3}}{\partial t}+iD_{3}\frac{\partial^{2}\bar{A }_{3}}{\partial t^{2}}+3i\gamma\bar{A}_{1}\bar{A}_{2}e^{i(\Delta_{31}k-\bar{A} _{21}k)z}=0,\,0<z\leq L_{z},0<t<L_{t},\]  
where  
\[\bar{\Delta}_{21}k =\Delta_{21}k+D_{1}\frac{v_{31}^{2}}{2(3D_{3}-D_{1})^{2}}-D_{2} \frac{v_{31}^{2}}{(3D_{3}-D_{1})^{2}}+\frac{\
u_{21}\
u_{31}}{3D_{3}-D_{1}},\] \[\bar{\Delta}_{31}k =\Delta_{31}k+\frac{3v_{31}^{2}}{4(3D_{3}-D_{1})}.\]  
So, if we choose the phase mismatching between TH and FW in appropriate way:  
\[\Delta_{31}k=-\frac{3v_{31}^{2}}{4(3D_{3}-D_{1})}, \tag{33}\]  
then the possible highest efficiency of the frequency tripling process is achieved. In turn, the phase mismatching as well as GVM between SH and FW also change according to the rules given by:  
\[ \begin{split}&\bar{v}_{21}=\
u_{21}-\frac{4D_{2}v_{31}}{6D_{3}-2D_{1} },\\ &\bar{\Delta}_{21}k=\Delta_{21}k+D_{1}\frac{\
u_{31}^{2}}{2(3D_{ 3}-D_{1})^{2}}-D_{2}\frac{\
u_{31}^{2}}{(3D_{3}-D_{1})^{2}}+\frac{\
u_{21} \
u_{31}}{3D_{3}-D_{1}}.\end{split}  \tag{34}\]  
We see from (33) that the phase mismatching \(\Delta_{31}k\), which is necessary to choose for the GVM compensation, does not depend on the incident pulse duration and is determined only by the difference between group velocities of the waves, and their SOD coefficients. However, it depends on the normalization length \(Z_{n}\) of the \(z\)-coordinate. The product of GVM and the length of normalization characterize the time shift between the centers of two pulses on this propagation distance. Now, let us estimate the parameters for GVM compensation. For definiteness, we choose the incident pulse duration being equal to 100 fs for KDP crystal.