## 2 Theory
### Tensor Response of Heating and Reorientation
#### 2.2.2 Angular Hole Burning  
Angular hole burning in an azo dye such as DR1 starts with the molecule changing length when excited by light. We view the trans molecule as being uniaxial along the 3 axis as shown in Figure 3. Since the trans molecule is long and thin, we assume its orientation to be fixed by steric interactions with the polymer host. The left-hand part of Figure 3 shows such a molecule. In the cis form, the molecule becomes more compact, so we assume that it can freely reorient. Its time-averaged shape then will be a sphere, as shown by the right-hand part of Figure 3. As such, the strain along the uniaxial direction, \(u_{33}\), is negative because of the decrease in its length while it is positive perpendicular to it. Figure 3 shows one molecule with its uniaxial axis tilted by an angle \(\theta\) relative to the lab frame's \(z\) axis. Here we review how changes in the isomer population induced by light polarized along \(z\) will change the population-averaged strain, which will lead to an anisotropic change in the material's shape. We will then use the fact that the strain is related to the stress through Young's modulus of the surrounding polymer, with which it interacts. For a molecule oriented at Euler angles \((\theta,\phi,\psi)\), the molecular strain in the principle axis frame 123 are given by \(u_{11}\), \(u_{22}\) and \(u_{33}\). These should be viewed as the strains that are induced by the light, which we will take into account shortly. The molecular strains are related to the strains \(u_{xx}\), \(u_{yy}\) and \(u_{zz}\) in the lab frame through  
\[ \begin{split} u_{zz}(\theta,\phi,\psi)&=a_{z1}( \theta,\phi,\psi)a_{z1}(\theta,\phi,\psi)u_{11}\\ &+a_{z2}(\theta,\phi,\psi)a_{z2}(\theta,\phi,\psi)u_{z2}\\ &+a_{z3}(\theta,\phi,\psi)a_{z3}(\theta,\phi,\psi)u_{33},\end{split}  \tag{22}\]where \(a_{z1}(\theta,\phi,\psi)\) is the \(z,1\) tensor component of the rotations matrix. A similar expression is found for \(u_{xx}\) and \(u_{yy}\). For a uniaxial molecule \(u_{22}=u_{11}\) and if the molecules are arranged to form a uniaxial material, then \(u_{yy}=u_{xx}\). A more compact way of expressing Equation (22) for all of the diagonal strain components is of the form  
\[u_{ii}(\theta,\phi,\psi)=\sum_{I=1}^{3}a_{II}(\theta,\phi,\psi)a_{II}(\theta, \phi,\psi)u_{II}, \tag{23}\]  
where \(I\) is the component in the molecular frame and \(i\) in the lab frame. Light polarized along the \(z\)-direction in the lab frame of magnitude \(E_{0}\) will have a component along the molecule's 3-axis of \(E_{0}\cos\theta\). We assume that the molecule absorbs light with an the electric field component along the 3-axis. The light-induced strain is proportional to the light energy absorbed and the absorbed energy is proportional to the square of the electric field component along the molecule's axis \(E_{0}^{2}\cos^{2}\theta\). The crucial part here is that the angular dependence of the light-induced strain is given by \(\cos^{2}\theta\). Putting it all together, Equation (23) becomes  
\[u_{ii}(\theta,\phi,\psi)=\sum_{I=1}^{3}a_{II}(\theta,\phi,\psi)a_{II}(\theta, \phi,\psi)u_{II}^{(0)}\cos^{2}\theta, \tag{24}\]  
where \(u_{II}^{(0)}\) is the light-induced strain for a molecule whose 3 axis is aligned with the \(z\) axis. Equation (24) has built into it the assumption that only light polarized along the long length of the trans isomer absorbs light, vanishing when \(\theta=\pi/2\). If the material is isotropic, the molecules are initially randomly oriented, so the averaged strain matrix in the lab frame is given by  
\[\bar{u}_{ii}(\theta,\phi,\psi) =\int d\Omega\,u_{ii}(\theta,\phi,\psi)\] \[=\int d\Omega\,\sum_{I=1}^{3}a_{II}(\theta,\phi,\psi)a_{II}( \theta,\phi,\psi)u_{II}^{(0)}\cos^{2}\theta, \tag{25}\]  
where we have used Equation (24). \(\Omega\) corresponds to the triplet of the three Euler angles and \(d\Omega\) is the triple integral over the three Euler angles. Details regarding the rotation matrix and evaluating such integrals can be found in the literature [51,54].