# Numerical Technique for Study of Noise Grating Dynamics in Holographic Photopolymers
## 4 Results
Conversely, when no diffusion occurs during writing, the haze growth is limited and shows no distinct patterns. This also serves to illustrate that haze growth in two-beam holography is distinctly different than self-written waveguides [3] in which scatter centers seed filaments whose scattering would be approximately isotropic in the transverse plane. Figure 7: Typical grating and haze growth versus exposure time in the weak regime for 10 wt% BPADA operating in the reaction-limited regime (\(R_{m}=400\)). Haze is observed to only grow significantly above the initial value of approximately 0.1 when the hologram efficiency is saturated, in this case in the over-modulated regime. The top and bottom plots of Figure 8 reveal how recording dynamics impact grating and scatter development. In the top diffusion-limited regime of \(R_{m}=0\), negligible haze has developed because the feedback process whereby noise gratings seed the growth of additional gratings has been suppressed. Simultaneously, the single strong hologram has been limited in its diffraction efficiency because monomer was unable to replenish that depleted in the bright regions of the fringe. Conversely in the bottom reaction-limited case of \(R_{m}\rightarrow\infty\), the experimentally observed scatter patterns (see e.g., Figure 1) are observed. Diffraction efficiency is in the over-modulated regime since nearly all the monomer has been consumed due to rapid transport into the bright fringes. A multiplexed set of weaker holograms with appropriate scheduling would yield the same total usable index contrast in both cases, but the upper case would exhibit greater SNR. The significant dependence of total haze upon the diffusion to reaction rate ratio shown in Figure 8 indicates \(R_{m}\) has a critical role in haze development. Figure 9 plots the final haze versus \(R_{m}\) under the same conditions as Figure 8. The plots show two distinct behaviors. When diffusion across a period is sufficiently slow that noise gratings cannot develop during the exposure, haze is limited to that which is recorded by the instantaneous index change. Conversely, when diffusion is sufficiently fast, noise gratings are self-reinforced during the exposure. The transition between these two regimes is seen to be approximately the boundary between diffusion-and reaction-limited systems at \(R_{m}\approx 1\). Critically, this indicates that the relevant spatial scale for scatter is approximately equal to the period, \(\Lambda\), consistent with the hypothesis that haze is not spatially random but is dominated by growth of Bragg  
Figure 8: Typical strong recording results for holographic (**left**) and haze (**right**) readouts in the case of slow (**top**) and fast (**bottom**) diffusion rate across a period relative to average polymerization rate. Only the lower case exhibits the scattering arcs and lines characteristic of Bragg-degenerate noise gratings. Intensities are plotted versus transverse spatial frequency in the material, rendered with a logarithmic color map to visualize the large dynamic range. Monomer loading in these 100 μm thick films is reduced from the previous plot to 10 wt% and initial haze is set to \(\approx 9\%\). Exposure time of \(\tau=2.5\) is sufficient to consume \(1-e^{-\tau}\approx 92\%\) of the monomer. degenerate noise gratings whose scale is similar to the grating period. This is contrast to the alternative hypothesis that the spatial scale of haze is similar to the particle size. Since scatter centers are typically much smaller than the period, noise dynamics controlled by diffusion would be much faster than signal dynamics in this case. The important implication for recording high SNR holograms is that both signal and noise obey similar dynamics such that diffusion-limited grating development will largely suppress haze development. The material and process variables that influence this condition are related by the single parameter \(R_{m}\).