## 2 Theory
### Tensor Response of Heating and Reorientation
#### 2.2.1 Heating and Molecular Reorientation  
At low light levels, \(\sigma=\sigma_{0}+\kappa_{\sigma}^{(1)}I\), so the stress is linear in the intensity and the ratio of the stresses for the two orthogonal pump polarizations for heating and molecular reorientation is given by [50]  
\[\frac{\sigma_{//}-\sigma_{0}}{\sigma_{\perp}-\sigma_{0}}=\frac{-\kappa_{//}^{(1 )}I}{-\kappa_{\perp}^{(1)}I}=\frac{\kappa_{//}^{(1)}(I)}{\kappa_{\perp}^{(1)} (I)}=\frac{\kappa_{//}^{(heat)}(I)+\kappa_{//}^{(re)}(I)}{\kappa_{\perp}^{( heat)}(I)+\kappa_{\perp}^{(re)}}, \tag{8}\]  
where \(\sigma_{//}\) is the stress along the sample's long axis and \(\sigma_{\perp}\) is perpendicular to it. \(\sigma_{0}\) is the pre-stress.  
Figure 2: After the hole is “burned” into the material as shown in Figure 1, the cis molecules decay back into the trans state with no preferred orientation. The net effect is that trans molecules oriented along the light’s polarization axis are converted into the orthogonal orientation.  
At higher intensities, where \(\sigma\) is no longer a linear function of the intensity, we can define \(\kappa_{\sigma}^{(1)}(I_{0})\) to be the slope of the stress curve at intensity \(I_{0}\), given by  
\[\kappa_{\sigma}^{(1)}(I_{0})=\left.\frac{\partial\sigma(I)}{\partial I}\right|_{ I_{0}}. \tag{9}\]  
\(\kappa_{\sigma}^{(1)}(I_{0})\) is then a material property that gives the photomechanical response for a small change in the intensity about a larger intensity \(I_{0}\). Alternatively, we can define \(\kappa_{\sigma}(I)\) as  
\[\kappa_{\sigma}(I)=\frac{\sigma(I)-\sigma(0)}{I}. \tag{10}\]  
Thus in the low intensity limit, the photomechanical response is linear and \(\kappa_{\sigma}^{(1)}\) is the linear photomechanical response. \(\kappa_{\sigma}^{(1)}(I_{0})\) is the intensity-dependent linear response that quantifies the change in stress for a small change in the intensity about \(I_{0}\), while \(\kappa_{\sigma}(I)\) describes the change in stress for large intensity \(I\) over a range where the intensity dependence is nonlinear. In the narrative that follows, we will not specify which definition is being used, but, generically, will show an intensity argument, such as \(\kappa^{(heat)}(I)\). We will revisit these concepts later.  
The photothermal heating contribution for an isotropic material is independent of polarization, so  
\[\kappa_{//}^{(heat)}(I)=\kappa_{\perp}^{(heat)}(I)\equiv\kappa^{(heat)}(I). \tag{11}\]  
For volume-preserving processes such as molecular reorientation, as the length of the sample decreases in one direction, it will increases in the other. If the orientational distribution function retains azimuthal symmetry, conservation of volume demands that  
\[\kappa_{//}^{(re)}(I)=-2\kappa_{\perp}^  
and stress parallel to the light's polarization is  
\[\kappa^{(1)}_{//}(I)=\kappa^{(heat)}_{//}(I)+\kappa^{(re)}_{//}(I),\] \[=(1+R^{(re)}(I))\kappa^{(heat)}(I) \tag{16}\]  
Therefore, \(\kappa^{(heat)}(I)\) and \(\kappa^{(re)}(I)\) can be derived using Equations (15) and (16), yielding  
\[\kappa^{(heat)}(I)=\frac{\kappa^{(1)}_{//}(I)}{1+R^{(re)}(I)}, \tag{17}\]  
and  
\[\kappa^{(re)}(I)=\frac{R^{(re)}\kappa^{(1)}_{//}(I)}{1+R^{(re)}(I)}. \tag{18}\]  
Finally, using Equations (11), (12), (17) and (18) the contribution of heating and molecular reorientation for each polarization of light is determined, yielding  
\[\kappa^{(heat)}_{//}(I)=\kappa^{(heat)}_{\perp}(I)=\kappa^{(heat)}(I)=\frac{ \kappa^{(1)}_{//}(I)}{1+R^{(re)}(I)}, \tag{19}\]  
\[\kappa^{(re)}_{//}(I)=\frac{R^{(re)}\kappa^{(1)}_{//}(I)}{1+R^{(re)}(I)}, \tag{20}\]  
and  
\[\kappa^{(re)}_{\perp}(I)=-\frac{1}{2}\frac{R^{(re)}\kappa^{(1)}_{//}(I)}{1+R^ {(re)}(I)}. \tag{21}\]