# Reversal of Photoinduced Bending of Crystals Due to Internal Refraction of Light
## 4 Model of the Interaction of Light with a Crystal
### Theoretical Formulation and Calculation Method  
To simulate wave propagation in a circular crystal, one can use the solution of the problem for the case of normal incidence of light on an infinite round cylinder, similar to the well-known Mie theory for light scattering by spherical particles [41; 42; 43]. The general solution is represented as a superposition of solutions corresponding to two different  
Figure 4: Dependences of the crystal curvature on the transformation time for different light sources. Figure 5: Qualitative scheme of light propagation in a crystal of round cross section. polarizations of the incident wave. In the case of \(s\)-wave (the electric field of the original wave is perpendicular to the plane of incidence) the electric field vector \(\mathbf{E}\) is parallel to the crystal axis in the whole space. In the case of \(p\)-wave (the electric field vector in the plane of incidence), the electric field of the initial wave is polarized perpendicular to the crystal axis, while the magnetic field vector \(\mathbf{H}\) is parallel to the crystal axis in the whole space. Further calculations using this model show that the intensity distributions in the crystal cross section obtained for different polarizations lead to qualitatively similar results for the relative intensity moment in Equation (7) in all cases considered. The insignificant quantitative difference is associated with the difference in the reflection coefficients for \(s\) and \(p\) waves at oblique incidence of the wave on the interface between the media [42]. Therefore, to shorten the discussed results, we will analyze only the \(s\)-wave case, which turns out to be less demanding on computational resources when solving the problem numerically. The complex amplitude of \(z\) component of the electric field for the \(s\)-wave case is determined by the following expressions in a cylindrical coordinate system (\(r\),\(\theta\)). The electric field inside the crystal \(E_{\text{int}}\) is defined by  
\[ \begin{split}& E_{\text{int}}(r,\theta)\ =\ E_{0}\sum\limits_{n\ =-\infty}^{\infty}F_{n}(\theta)d_{n}J_{n}(mk_{0}r)\\ & F_{n}(\theta)\ =\ (-i)^{n}e^{in\theta}\\ & d_{n}\ =\ \frac{J_{n^{\prime}}(k_{0}\theta)\ H_{n}(k_{0}\theta)-J_{n}(k_{0} \theta)\ H_{n^{\prime}}(k_{0}\theta)}{mJ_{n^{\prime}}(mk_{0}\theta)\ H_{n}(k_{0} \theta)-J_{n}(mk_{0}\theta)\ H_{n^{\prime}}(k_{0}\theta)}\end{split}  \tag{8}\]  
where \(E_{0}\) is the amplitude of the plane wave normally incident on the crystal in Figure 5;  
\[E_{\text{inc}}\ =\ E_{0}e^{i\hat{k}_{0}x}, \tag{9}\]  
characterized by the wave vector \(k_{0}\) = 2\(\pi\)/\(\lambda\) and wavelength \(\lambda\) in vacuum, \(m\) is the complex refractive index of the crystal (the crystal is considered non-magnetic);  
\[m\ =\ \sqrt{\epsilon^{\prime}+i\epsilon^{\prime\prime}}, \tag{10}\]  
determined by the real and imaginary components, \(\epsilon^{\prime}\ \
\ \epsilon^{\prime\prime}\), of the relative complex permittivity \(\epsilon=\epsilon^{\prime}+i\epsilon^{\prime\prime}\), \(J_{n}\) and \(H_{n}\) are the Bessel and Hankel functions of the first kind of order \(n\), primes denote the derivative. The field of the scattered wave \(E_{\text{s}}\) is determined by Equation (11);  
\[ \begin{split}& E_{\text{s}}(r,\theta)\ =\ -E_{0}\sum\limits_{n\ =-\infty}^{\infty}F_{n}(\theta)b_{n}H_{n}(k_{0}r)\\ & b_{n}\ =\ \frac{mJ_{n^{\prime}}(mk_{0}\theta)I_{n}(k_{0}\theta)-I_{n}(mk_{0} \theta)I_{n^{\prime}}(k_{0}\theta)}{mJ_{n^{\prime}}(mk_{0}\theta)H_{n}(k_{0} \theta)-I_{n}(mk_{0}\theta)H_{n^{\prime}}(k_{0}\theta)}\end{split}  \tag{11}\]  
To calculate the field, we used the constant value of the real component of relative permittivity \(\epsilon^{\prime}=\epsilon_{0}=2.5\), which corresponds to the refractive index \(n_{0}=\epsilon_{0}^{1/2}\)-\(1.58\), typical for inorganic glasses.