# Numerical Technique for Study of Noise Grating Dynamics in Holographic Photopolymers
## 3 Validation  
To ensure the appropriateness and proper implementation of this combination of methods, it is critical that the code suite be validated. There are limited theoretical methods for predicting scatter development in these materials and the problem is inherently statistical. Thus, we instead validate the technique using a fully theoretical model of two-beam Bragg holography. A comparison to theory is chosen rather than experiment to enable direct comparison with no fitting to find parameters that are not always easily accessible experimentally, e.g., diffusion and reaction rates. Combining Equations (2) and (4) results in a single rate equation governing this simplified system. Restricting the analysis to one dimension in the plane of incidence gives:  
\[\frac{d}{dt}[m(x,t)]=D\
abla^{2}[m(x,t)]-F_{P}[1+V_{m^{*}}\cos{(k\,x)}][m(x,t)], \tag{7}\]  
where \(F_{P}=k_{P}k_{I}\tilde{1}/k_{T}\) is the average polymerization rate. Equation (7) can be normalized with the substitutions:  
\[\tau=t\,F_{P},\;\chi=x/\Lambda,\;m=\frac{[m]}{[m_{0}]},\;R_{m}=\frac{D\,k^{2}} {F_{P}}, \tag{8}\]  
where \([m_{0}]\) is the formulated monomer concentration. This transforms Equation (5) into a unitless form:  
\[\frac{\partial}{\partial\tau}m(\chi,\tau)=\frac{R_{m}}{(2\pi)^{2}}\frac{ \partial^{2}}{\partial\chi^{2}}m(\chi,\tau)-[1+V_{m^{*}}\cos{(2\,\pi\chi)}]\;m( \chi,\tau), \tag{9}\]  
which admits a standard harmonic solution \(m(\chi,\tau)=\sum_{i=0}^{4}m_{i}(\tau)\;\cos{(2\,\pi\;i\;\chi)}\), here expanded up to 4 th order [12]. The resulting set of coupled first-order ordinary differential equations is:  
\[ \begin{array}{c}\frac{\partial}{\partial\tau}m_{0}(\tau)=-\;m_{0}(\tau)- \frac{V_{m^{*}}}{2}m_{1}(\tau)\;,\\ \frac{\partial}{\partial\tau}m_{1}(\tau)=-\;V_{m^{*}}\;m_{0}(\tau)-\;(1+R_{m} )m_{1}(\tau)-\frac{V_{m^{*}}}{2}m_{2}(\tau),\\ \frac{\partial}{\partial\tau}m_{1}(\tau)=-\;V_{m^{*}}\;m_{0}(\tau)-\;(1+R_{m} )m_{1}(\tau)-\frac{V_{m^{*}}}{2}m_{2}(\tau),\\ \frac{\partial}{\partial\tau}m_{2}(\tau)=-\frac{V_{m^{*}}}{2}m_{1}(\tau)-(1+4 \;R_{m})m_{2}(\tau)-\frac{V_{m^{*}}}{2}m_{3}(\tau),\\ \frac{\partial}{\partial\tau}m_{3}(\tau)=-\frac{V_{m^{*}}}{2}\;m_{2}(\tau)-(1+ 9\;R_{m})m_{3}(\tau),\\ \frac{\partial}{\partial\tau}n_{4}(\tau)=-\frac{V_{m^{*}}}{2}\;m_{3}(\tau). \end{array}  \tag{10}\]  
These equations describe how the sinusoidal radical distribution of finite visibility given by \(1+V_{m^{*}}\cos{(2\,\pi\chi)}\) drives the evolution of monomer concentration as a sum of grating harmonics. Consider the ideal recording case where diffusion is much faster than propagation such that \(R_{m}\rightarrow\infty\). The equations show that all harmonics of the monomer higher than zero would remain identically zero, consistent with fast diffusion enforcing a spatially uniform monomer distribution. The average monomer concentration, \(m_{0}(\tau)\), would then decay exponentially like \(\exp{(-\tau)}\) which is the ideal first order reaction with time constant \(\tau=t\,F_{P}\) where \(F_{P}\) is the average rate of polymerization. Equation (10) is a sufficiently simple set of coupled first-order differential equations that it can be solved in closed form with the help of a symbolic mathematics package. With the monomer distribution \(m(\chi,\tau)\) known, the polymer distribution can then be found from the time integral of the last term in Equation (9):  
\[P(\chi,\tau)=\left[1+V_{m^{\prime}}\cos\left(k\;\chi\right)\right]\int\limits_{ 0}^{\tau}m(\chi,\tau^{\prime})\;d\tau^{\prime}, \tag{11}\]  
where \(P(\chi,\tau)\) is the polymer concentration normalized to \(\left[m_{0}\right]\). \(P(\chi,\tau)\) represents the local concentration of monomer units that have reacted to become polymer. The left-hand side of Equation (11) is harmonically expanded as \(P(\chi,\tau)=\sum_{i=0}^{4}p_{i}(\tau)\;\cos\left(2\;\pi\;i\;\chi\right)\) and terms for each harmonic \(\cos\left(2\;\pi\;i\;\chi\right)\) are set equal, generating integral equations for each polymer harmonic \(p_{i}(\tau)\).