# Reversal of Photoinduced Bending of Crystals Due to Internal Refraction of Light
## 4 Model of the Interaction of Light with a Crystal
### Simulation Results  
#### 4.3.1 Round Cross Section  
Figure 6 visualizes the relative radiation intensity \(I/I_{0}\), calculated for the wave defined in Equations (8) and (11) for two cases that reproduce the experimental conditions of irradiation of the 2 \(\upmu\)m thick crystal by the light sources \(\lambda\) = 400 nm (absorption depth \(\mu^{-1}\) = 30 \(\upmu\)m) and \(\lambda\) = 350 nm (\(\mu^{-1}\) = 1 \(\upmu\)m). The internal oscillating structure of the intensity is associated with the interference of internal reflected waves: the maxima and minima of the intensity alternate with the period of half the wavelength in the substance, \(\lambda/(2e_{0}^{1/2})\). In addition, the resulting interference pattern gives an idea of the shape of the wave fronts inside the crystal, visualizing the condensation of light as a result of refraction. Strictly speaking, such an interference pattern is valid for a monochromatic source. However, with a typical spectrum width of the used sources \(\Delta\lambda\)-0.03\(\lambda\)[39] the characteristic coherence length is \(\sim\)30\(\lambda\). That is, the calculation result for a monochromatic source quite correctly shows the real intensity distribution for crystals thinner than 10 \(\upmu\)m in the case of using light sources like LED for irradiation. This means that in addition to the effects discussed in this study, sufficiently thin crystals should also contain a fine structure of the transformation rate. Figure 7 shows 3D surfaces of \(I/I_{0}\) inside the crystal for these two cases. As can be seen, in the case of weak absorption (\(\lambda\) = 400 nm, Figure 7a) there is a tendency for a significant increase in the intensity towards the rear surface of the crystal, associated with the condensation of the light flux. For the case of strong absorption (\(\lambda\) = 350 nm, Figure 7b), there is a regular decrease in the intensity deep into the crystal. For a clearer idea of how the obtained distributions form the intensity moments of different signs, one can consider the dependences of the radiation power incident on the flat layers of matter inside the crystal (layers shown in blue and red in Figure 5), depending on the distance to the neutral surface \(x\). The value of the relative power, normalized to the power incident on the entire crystal, is determined by the intensity integral over the layer height, Equation (18);  
\[\frac{P(x)}{P_{0}}\ =\ \frac{1}{h}\int\limits_{\Delta y}\frac{I(x,y)}{I_{0}}dy. \tag{18}\]  
Relative power graphs are shown in Figure 8. In the case of weak absorption, the power increases with \(x\) in the front half of the crystal and saturates in the back half, as was qualitatively predicted above, which leads to a faster conversion on the back side of the crystal. In the case of strong absorption, the radiation power maximum is localized near the front surface of the crystal, as is usually assumed in all works studying the photo  
Figure 6: Visualization of the intensity distribution of a wave scattered by a 2 \(\upmu\)m thick crystal for two cases considered in the experiment: (**a**) \(\lambda\) = 400 nm, \(\mu^{-1}\) = 30 \(\upmu\)m, (**b**) \(\lambda\) = 350 nm, \(\mu^{-1}\) = 1 \(\upmu\)m. The arrows show the direction of the incident light. Figure 7: 3D light intensity distributions inside the crystal for the two cases shown in Figure 6: (**a**) \(\lambda\) = 400 nm, \(\mu^{-1}\) = 30 \(\upmu\)m, (**b**) \(\lambda\) = 350 nm, \(\mu^{-1}\) = 1 \(\upmu\)m.  
Figure 9 shows the results of calculations of the reduced moment of intensity \(\Omega_{r}\)  
\[\Omega_{r}\ =\ \frac{\Omega_{0}}{\varepsilon_{0}I_{0}\varphi\mu\
u}\frac{J}{h^{3}} \ =\ -\frac{1}{h^{3}}\int\limits_{S}x\,\frac{I(x,y)}{I_{0}}ds, \tag{19}\]  
a dimensionless characteristic, convenient for comparing the results depending on the product \(\mu h\), which characterizes the absorption of light by a crystal with a diameter \(h\).