## 3 Results and Discussion
### Parameter Estimation  
Taking into account the experimental results of the previous two sections our aim was to obtain some basic parameters of the biophotopol compositions for thin layers (around 100 \(\mu\)m thick). Shrinkage is due to layer compaction during polymerization. To determine the velocity of the chemical reaction, the polymerization rate, FR, we used Equation (13) presented in [13]. The FR results for each composition are shown in Table 3. The correlation value of the fittings indicates that there is good agreement between theory and experiments. In addition, compositions A and D have similar values of  
Figure 4: Angular scan for the holograms recorded in the 900 \(\mu\)m photopolymer layers. FR because the only difference between them is the monomer concentration. We have omitted the fitting for composition F due to the poor repeatability. The problem with the 100 um layers is that the solubility of PRF inside the layer depends dramatically on the drying conditions. The values for \(n_{p}\) are important when compared with the monomer values; _i.e._, the refractive index of acrylamide is clearly lower than that of NaAO. To determine these values we followed the steps indicated in [17]. Furthermore, the most important point is to compare \(n_{m}\) with \(n_{p}\) and these parameters with \(n_{b}\). In this sense the importance of a crosslinker to increase \(n_{p}\) was demonstrated using the zero frequency technique. The low value obtained for composition C can be explained by the precipitation of part of the BMA during the drying process. To improve this composition the concentration of TEA should be increased as in composition D and the compositions described in Ref. [17]. As can be seen in Table 4, the value of \(n_{p}\) for composition A is lower than for composition D. This can be explained by the higher concentration of monomer, which leads to longer polymer chains (larger values of refractive index). Once the chemical parameters have been determined, the monomer diffusion can be fitted recording very low spatial frequency gratings. To determine the NaAO diffusion inside the material we measured the first eight diffracted orders (reflected in this case) during recording (30 s) and the dark evolution of the diffraction efficiencies after recording. Figure 5 shows the behaviour for a recording of spatial period 0.168 um and the post-exposure evolution. Following the steps described in [18] we obtained the surface profile of the layer by fitting the experimental DE data with a Fermi-Dirac function. It is well known that during recording the illuminated photopolymer zones shrink due to polymerization and subsequently swell due to monomer diffusion from the dark zones. The velocity of swelling depends on the material diffusivity, \(D\), the diffusion time, \(\tau\), and the grating period, \(\Lambda\):  
\[D=\frac{\Lambda^{2}}{4\pi^{2}\tau} \tag{1}\]  
Using the data from Figure 5 we obtain the following values: \(\tau\)= 5000 s and \(D=1.4\times 10^{-9}\) cm\({}^{2}\) s\({}^{-1}\). Once the polymerization rates and monomer diffusion inside the material are known, we can apply the holographic diffusion models to simulate hologram formation in these photopolymers [19]. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Composition A** & **Composition B** & **Composition C** & **Composition D** & **Composition E** \\ \hline Initial monomer & & & & & \\ volume fraction & 0.13 & 0.15 & 0.25 & 0.17 & 0.11 \\ \(n_{p}\) & 1.578 & 1.603 & 1.585 & 1.595 & 1.51 \\ \(n_{b}\) & 1.470 & 1.476 & 1.476 & 1.474 & 1.470 \\ \hline \end{tabular}
\end{table} 
Table 4: Initial monomer volume fraction, polymer refractive index, \(n_{p}\), and binder refractive index, \(n_{b}\), for each composition based on Biophotopol. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Composition A** & **Composition B** & **Composition C** & **Composition D** \\ \hline \(F_{R}\) x 10\({}^{-3}\) (s\({}^{-1}\)) & 3.2 & 3.6 & 4.1 & 3.3 \\ \(R^{2}\) & 0.999 & 0.993 & 0.992 & 0.994 \\ \hline \end{tabular}
\end{table}