# Numerical Technique for Study of Noise Grating Dynamics in Holographic Photopolymers
## 4 Results
Since haze is seeded by scattering, an obvious strategy for reduction of haze is to reduce the initial scatter, e.g., via resin filtering and clean-room processing. Given that recorded haze is an amplified growth process, it is possible that reductions in initial haze could have super-linear impact on final haze. Figure 10 shows how recorded diffraction efficiency and final haze scale with initial haze, controlled here via the cross section of Rayleigh scatter centers of the same number density of \(10^{-3}/\upmu\text{m}^{3}\). These results show that final haze is a fixed multiple of the initial haze. The ratio depends on the material and recording conditions. This result indicates that clean sample preparation is important, but has only linear impact on the final haze value. Figure 10: Final holographic efficiency (circles) and haze (squares) as a function of the initial haze. Haze is due to Rayleigh scattering for diffusion-limited (**left**) and reaction-limited (**right**) recording for the same material as Figures 8 and 9. Final haze is seen to be a fixed multiple of initial haze if the signal and noise gratings are not competing for monomer, that is haze plus diffraction efficiency is much less than one. The dependence of haze growth on the relative rate of diffusion and propagation reaction is consistent with Figure 9.  
Figure 9: Haze development as a function of \(R_{m}\), the ratio of diffusion rate across a period to average polymerization reaction rate for hologram period \(\Lambda=1\upmu\)m (**left**) and \(\Lambda=2\upmu\)m (**right**). The semi-logarithmic sigmoidal fit has an inflection point at \(R_{m}=2.5\)(**left**) and \(R_{m}=1.6\)(**right**), consistent with the hypothesis that haze development depends strongly on recording in the diffusion-limited (\(R_{m}\ll 1\)) or reaction-limited (\(R_{m}\gg 1\)) regime. The material is the same as in the previous figure. Finally, we consider the dependence of recorded haze on sample thickness. It is commonly observed experimentally that haze grows rapidly with sample thickness and this is confirmed in Figure 11 where the initial haze has been subtracted to highlight the dynamics of haze growth caused by the medium. The left and right plots illustrate how uniformly distributed scatter (left) produces less haze than the same scatter concentrated on the substrate through which the light enters (right). Conversely, the distributed scatter (left) exhibits greater nonlinearity versus thickness, consistent with noise gratings being written by scattered intensity which grows linearly in depth (left) but is initially constant in depth (right). While we have fit this data to cubic polynomials to guide the eye, the thickness dependence of the haze does not appear to have any simple scaling relationship. This may be due to the complexity of the writing intensity pattern for haze which is generated by scatter from the static and dynamic haze interfering with the reference and object beams whose intensity varies in depth as they couple. Once significant monomer has been consumed (\(\tau>0.5\)), the haze and the hologram also compete for monomer, slowing haze growth. This highlights the fact that complete understanding of haze dynamics requires a theoretical treatment. The numerical results and approach presented here may provide input to such an approach.