# Numerical Technique for Study of Noise Grating Dynamics in Holographic Photopolymers
## 4 Results  
The validated code is now applied to investigate how noise and signal develop in a holographic photopolymer containing scatter centers. The holographic photopolymer uses bisphenol A diacrylate (BPADA) as a writing monomer in a polyurethane matrix initiated with 2,4,6-trimethylbenzoylyl diphenylphosphine oxide (TPO) at 405 nm. At 633 nm, the refractive indices of the monomer, polymer and matrix are 1.545, 1.5789, and 1.476 respectively. Dispersion of the indices is not included in this study. The shrinkage of the monomer in matrix upon full polymerization is found via the slanted grating technique [15] to be \(\sigma=0.14\), consistent with typical vales of 22-24 mL/mol for complete shrinkage of acrylate monomers. Initiator concentration is sufficiently low that the samples may be considered optically thin and, thus, no absorption is included. Rather than use measured diffusivity, reaction rates, initiator concentration and intensity to calculate a specific value for the diffusion to reaction rate ratios for monomer and radicals, \(R_{m}\) and \(R_{m^{\prime}}\), we will take these to be free variables in order to explore their role in haze development. Note that these ratios are not fixed even for a specific material since they depend on hologram period and optical intensity, thus, they can be varied  
Figure 6: Solid line: Theoretical calculation of Bragg selectivity of a 10 μm thick sample of 20 wt% bisphenol A diacrylate in a solid polyurethane matrix illuminated with Gaussian beams at 405 nm to form a one μm period hologram with \(1/e^{2}\) field diameter of 200 μm. The peak intensity is chosen to operate in the diffusion-limited regime where higher-order fringe harmonics are expected (\(R_{m}=0.2\)). The exposure time is 2.5 exponential time constants of the mean polymerization rate, that is the monomer concentration is reduced to \(e^{-2.5}\) of its initial value, averaged over a holographic fringe. Circles: The same calculation performed by the numerical code described in the Methods section. The reference probe beam has a \(1/e^{2}\) intensity diameter of 40 μm. This reduced efficiency due to the overlap of the Gaussian hologram shape and the Gaussian read beam has been corrected as described in [14]. experimentally to exploit the trends shown next. Similarly, we take initial haze to be a free variable to understand its importance in haze development. The writing beams are tilted, except where stated, to produce a grating period of one \(\upmu\)m and have a 1/\(e^{2}\) intensity diameter of 200 \(\upmu\)m. The readout beam used for nondestructive measurement of diffraction efficiency and haze is at 660 nm with a 40 \(\upmu\)m 1/\(e^{2}\) intensity diameter, tilted at the appropriate Bragg matched angle or normally incident, respectively. Initial haze is generated by Rayleigh scatter centers with number density \(10^{-3}/\upmu\)m\({}^{3}\) whose cross-section is adjusted to give the desired initial haze. Typical growth dynamics are shown in Figure 7. Even for this case with relatively high initial haze of nearly 0.1, haze is observed to grow much more slowly than diffraction efficiency. In this case, the hologram is overmodulated before significant haze growth occurs. This observation is consistent with the hypothesis that noise gratings are written by interference of writing beams and scattered light while the intended grating is written by the much stronger interference of reference and object beams. The first result of this numerical technique is therefore that significant haze growth arising from amplification of Bragg-matched noise gratings only occurs for holograms that are strongly saturated. Haze growth via amplification of Bragg-matched noise gratings is further supported by recording in diffusion and reaction-limited materials as shown in Figure 8. The distinct patterns visible in the spatial frequency spectrum of the scatter in the lower plots are consistent with Bragg degenerate noise gratings amplified during writing by fast diffusion.