## 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
So, SOD parameters are \(D_{1}=0.0006\), \(D_{2}=-0.0035\), \(D_{3}=-0.0062\), and the GVM parameters are \(\
u_{21}=0.67\), \(\
u_{31}=1.02\). The phase mismatching between SH and FW equals 127. According to the provided analysis, the initial phase modulation must be chosen as follows:  
\[\varphi(t)=\frac{\
u_{31}}{6D_{3}-2D_{1}}t=-26.5625t.\]  
To compensate the phase mismatching induced by GVM, we choose the phase mismatching between TH and FW as follows:  
\[\Delta_{31}k=-\frac{3v_{31}^{2}}{4(3D_{3}-D_{1})}=40.640625.\]For practice, it is important to compare this frequency shifting with FW carrier frequency. Let us write the complex amplitude in physical variables:  
\[\bar{A}_{1}(z,\bar{t})=\sqrt{I_{01}}e^{-\frac{(\bar{t}-0S_{1}t)^{2}}{\tau_{p}^{2} }}e^{-i(2\pi(v+\Delta v)\bar{t}-k\bar{z})}, \tag{35}\]  
where \(\bar{A}_{1}\), \(\bar{z}\), \(\bar{t}\), \(\tau_{p}\), \(\bar{k}\) are introduced in (3), and \(\
u\) is the FW carrier frequency. For a 1064 nm wavelength, it is approximately equal to 300 THz. \(\Delta\
u\) is the frequency shift. The dimensionless value of \(2\pi v\bar{t}\) is defined as follows:  
\[2\pi v\bar{t}=2\pi v\tau_{p}t\approx 186t.\]  
which means that the frequency shift \(\Delta\
u\) is about 14.2% of the FW carrier frequency \(\
u\), and may be reached in some cases, as we believe. It is also should be stressed that the ratio \(\Delta\
u\):\(\
u\) does not depend on the pulse duration \(\tau_{p}\):  
\[\frac{\Delta\
u}{\
u}=-\frac{\left(\frac{\Delta\bar{t}}{\partial\bar{\omega} }\right|_{3\bar{\omega}}-\frac{\partial\bar{\omega}}{\partial\bar{\omega}} \left|{}_{\bar{\omega}}\right)}{4\pi\left(3\frac{\partial^{2}\bar{k}}{ \partial\bar{\omega}^{2}}\left|{}_{3\bar{\omega}}-\frac{\partial^{2}\bar{k}}{ \partial\bar{\omega}^{2}}\left|{}_{\bar{\omega}}\right)\
u\right.}.\]  
To verify the theoretical analysis, we compare the computer simulation results obtained by using the equation set (1) with the corresponding results obtained by using the equation set (32). For this aim, we also have to find the value of the modified GVM coefficient between SH and FW:  
\[\
u_{21}-\frac{4D_{2}\
u_{31}}{6D_{3}-2D_{1}}=0.298125.\]  
Therefore, the value of the modified phase mismatching between SH and FW is chosen as follows:  
\[\bar{\Delta}_{21}k=\Delta_{21}k-4D_{2}\frac{\
u_{31}^{2}}{(6D_{3}-2D_{1})^{2 }}+\frac{2\
u_{21}\
u_{31}}{6D_{3}-2D_{1}}+2D_{1}\frac{\
u_{31}^{2}}{(6D_{3}- 2D_{1})^{2}}=102.13085975.\]  
The corresponding results are shown in Figure 25.  
Figure 25: Comparison of the pulse shapes in the section \(z=70\) (**a**). The wave intensities evolution (**b**), as well as the frequency conversion efficiency (**c**), along \(z\)-coordinate for the solutions of the problem (1) at \(\gamma=1\), \(D_{1}=0.0006\), \(D_{2}=-0.0035\), \(D_{3}=-0.0062\) and \((\Delta_{21}k\), \(\Delta_{31}k\), \(\
u_{21}\), \(\
u_{31}\), \(\varphi(t))=(127\), 40.640625, 0.67, 1.02, \(-26.5625)\) (solid lines) and \((102.13085875,0,0.298125,0,0)\) (32) with initial conditions (2) (dotted lines). Numbers 1 and 3 and red and green lines denote FW and TH, respectively. First of all, let us note that two solutions of the problem (1) coincide with each other. This means that our theoretical consideration is correct. Secondly (and most importantly), the THG efficiency is high and reaches a value of about 48%, which is comparable with the efficiency obtained in Section 5 (for example, Figure 3). Of course, in the general case, the necessary frequency shift of the FW may not always be reached in practice. To our knowledge, the wavelength shift of the FW may be about 20  nm without losses of the conversion efficiency, if it is necessary to compensate the GVM influence. Therefore, we propose the following way to obtain the high-efficiency THG in one crystal. First of all, one can introduce the frequency shift. Then, by changing of an angle between the crystal axis and the pulse propagation direction, one may choose the optimal phase mismatching condition.