## 6 Theoretical Interpretation  
The disappearance of the waveguide is observed only when the sample is heated. An increase in the refractive index throughout the entire crystal volume occurs with an increase in the crystal temperature and leads to the waveguide contrast \(\Delta n=n_{2}-n_{1}\) compression, as well as to the termination of the channeling of the fundamental radiation mode (when the limiting contrast value is reached). The critical value \(\Delta n_{crit}\) can be estimated as follows:  
For simplicity, let us consider the propagation of a transverse-electric wave (TE-wave) with a wavelength \(\lambda\) along a planar waveguide with a step-graded index (Figure 7). The light propagation in the waveguide is described by single equation for a TE-wave (1):  
\[\frac{\partial^{2}E_{y}(x)}{\partial x^{2}}+(\frac{n^{2}\omega^{2}}{c^{2}}- \beta^{2})E_{y}(x)\ =\ 0, \tag{1}\]  
For the region \(x>0\) in Figure 7a, the solution to Equation (1) is:  
\[E_{y}(x)\ =\ E_{3}e^{-\gamma_{3}x}, \tag{2}\]  
where \(\gamma_{3}\ =\ \frac{\omega}{c}\sqrt{N^{2}-n_{3}^{2}}\). For the region \(-d<x<0\) in Figure 7a, the solution to Equation (1) is:  
\[E_{y}(x)\ =\ E_{2}\cos(k_{x}x+\varphi_{2}), \tag{3}\]  
Figure 7: A waveguide model with a step-graded index: (**a**) a refractive index profile in a planar waveguide; (**b**) the scheme of light propagation within a waveguide in the ray approximation. \(n_{1}=2.20\), \(n_{2}=2.21\), \(n_{3}=1\) are the refractive indices of the substrate, waveguide and coating (respectively), \(N\) is the effective refractive index, \(d\) is the thickness of the waveguide and \(\theta\) is the angle of incidence of the beam on the interface. where \(k_{x}\ =\ \frac{\omega}{c}\sqrt{n_{2}^{2}-N^{2}}\). For the region \(x<-d\) in Figure 7a, the solution to Equation (1) is:  
\[E_{y}(x)\ =\ E_{1}e^{-\gamma_{1}(x+d)}, \tag{4}\]  
where \(\gamma_{1}\ =\ \frac{\omega}{c}\sqrt{N^{2}-n_{1}^{2}}\). Based on the boundary conditions on the media interfaces \(n_{3}\) and \(n_{2}\), \(n_{1}\) and \(n_{2}\):  
\[ \begin{array}{rcl}E_{3}&=&E_{2}\cos(\varphi_{3})\\ \gamma_{3}E_{3}&=&k_{x}E_{2}\sin(\varphi_{3})\end{array} ,\quad \begin{array}{rcl}E_ {1}&=&E_{2}\cos(-k_{x}d+\varphi_{3})\\ \gamma_{1}E_{1}&=&-k_{x}E_{2}\sin(-k_{x}d+\varphi_{3})\end{array} , \tag{5}\]  
By dividing the lower equation from Equation (5) by the upper equation, we obtain the following system:  
\[ \begin{array}{rcl}\tan\varphi_{3}&=&\frac{\gamma_{3}}{k_{x}}\\ \tan(k_{x}d-\varphi_{3})&=&\frac{\gamma_{1}}{k_{x}}\end{array} , \tag{6}\]  
If we express the arguments of tangents in Equation (6) and substitute the upper equation into the lower one, we obtain the dispersion equation:  
\[k_{x}d\ =\ \tan^{-1}\!\left(\frac{\gamma_{1}}{k_{x}}\right)+\tan^{-1}\!\left( \frac{\gamma_{3}}{k_{x}}\right)+m\pi, \tag{7}\]  
By substituting \(k_{x}\), \(\gamma_{1}\) and \(\gamma_{1}\) in Equation (7), we obtain the following equations:  
\[\frac{2\pi}{\lambda}d\sqrt{n_{2}^{2}-N^{2}}\ =\ \tan^{-1}\!\left(\sqrt{\frac{N^{2}-n_{ 1}^{2}}{n_{2}^{2}-N^{2}}}\right)+\tan^{-1}\!\left(\sqrt{\frac{N^{2}-n_{3}^{2}} {n_{2}^{2}-N^{2}}}\right)+m\pi, \tag{8}\]  
By making substitutions \(V\equiv\frac{2\pi}{\lambda}d\sqrt{n_{2}^{2}-n_{1}^{2}}\), \(b_{E}\equiv\frac{N^{2}-n_{1}^{2}}{n_{2}^{2}-n_{1}^{2}}\), \(a_{E}\equiv\frac{n_{1}^{2}-n_{3}^{2}}{n_{2}^{2}-n_{1}^{2}}\), we obtain the normalized dispersion equation:  
\[V\sqrt{1-b_{E}}\ =\ \tan^{-1}\!\left(\sqrt{\frac{b_{E}}{1-b_{E}}}\right)+ \tan^{-1}\!\left(\sqrt{\frac{a_{E}+b_{E}}{1-b_{E}}}\right)+m\pi, \tag{9}\]  
In the case of a symmetric waveguide (\(n_{1}=n_{3}\), \(a_{E}=0\)), Equation (9) always has a solution, at least for the value \(m=0\). In the case of an asymmetric waveguide (\(n_{1}\
eq n_{3}\), \(a_{E}\
eq 0\)), not only the solutions for positive \(m\), but also the solutions for \(m=0\) successively disappear with decreasing \(\Delta n=n_{2}-n_{3}\) at a fixed layer thickness \(d\). Physically, this phenomenon can be explained by the fact that the fundamental mode begins to radiate into the substrate.