# Numerical Technique for Study of Noise Grating Dynamics in Holographic Photopolymers
## 2 Methods
### Photochemical Reactions  
This intensity then drives a sequence of chemical reactions, calculated through time-stepping the first order chemical rate equations. The first reaction in the sequence is the photolysis of an initiator (assumed to be present in sufficient excess that it does not significantly deplete) by the optical intensity to generate primary radicals with rate constant \(k_{l}\), which rapidly react to form a radical monomer \(m^{*}\). These radicals diffusive with diffusivity \(D\) and terminate via a unimolecular reaction  
Figure 2: Slice at y = 0 through the three-dimensional numerical space illustrating the numerical method employed. An initial index distribution of randomly distributed scatter centers (**a**) is created to seed the formation of noise gratings. This is illuminated from the left with two tilted coherent Gaussian beams that interfere as they propagate in thickness, z, forming the intensity pattern (**b**). The downward sloped red lines (top and bottom) indicate the 1/e waist of the object beam electric field while the upward tilted red lines (top and bottom) indicate the 1/e waist of the reference beam electric field. The upward tilted red lines (center) indicate the 1/e waist of the probe field that will be used to read out the eventual hologram. The interference of object and reference fields drives the photochemical reaction, diffusion and index-formation process, resulting in an index distribution (**c**) consisting of the hologram, haze and the static scatter centers. The visible evolution of the fringes from low to high noise as a function of depth illustrates the interplay between these structures within the material. with rate constant \(k_{T}\). We chose a unimolecular termination reaction to match the intensity reciprocity observed for two-component holographic photopolymers [8]. To our knowledge, the linear and reciprocal intensity response of these materials has not been adequately explained in the literature and is surprising given the otherwise universal sublinear intensity response of radically initiated photopolymers. With these assumptions, the radical concentration evolves as:  
\[\frac{d}{dt}[m^{*}(\overrightarrow{r},t)]=D\
abla^{2}[m^{*}(\overrightarrow{r },t)]-k_{T}[m^{*}(\overrightarrow{r},t)]+k_{I}(\overrightarrow{r},t). \tag{1}\]  
The termination rate constant \(k_{T}\) is assumed to encompass any first order process which prevents radicals from participating in the subsequent propagation reaction. In steady state, the solution to this equation under sinusoidal illumination, \(I(x)=\tilde{I}\left[1+\cos{(k\,x)}\right]\), is found to be a sinusoid with finite visibility:  
\[[m^{*}(\overrightarrow{r},t)]=\frac{k_{I}}{k_{T}}\tilde{I}\left[1+V_{m^{*}} \cos{(k\,x)}\right], \tag{2}\]  
where \(\tilde{I}\) is the average intensity, \(k\equiv 2\pi/\Lambda\) is the grating wave number and the radical visibility is given by:  
\[V_{m^{*}}\equiv\frac{1}{R_{m^{*}}+1}. \tag{3}\]  
where \(R_{m^{*}}\equiv Dk^{2}/k_{T}\) is the diffusion rate of radicals across a period relative to radical termination rate. As shown in Figure 3, the distribution of radicals is identical to the sinusoidal intensity (\(V_{m^{*}}=1\)) when this unitless ratio is much smaller than 1. However, when diffusion is sufficiently fast that radicals can diffuse into the dark region of the fringe before terminating (\(R_{m^{*}}\gg 1\)), the visibility of the radical distribution decreases inversely proportional to period squared. This behavior has previously been identified as the resolution limit in published materials [9]. This explains the observation that significant improvement in dynamic range can be achieved by increasing the termination rate via radical traps on the matrix [10]. Here we will assume the steady-state distribution of radicals with finite visibility given by Equation (3), governed by the unitless diffusion rate to termination rate ratio \(R_{m^{*}}\). Figure 3: Radical fringe visibility, \(V_{m^{*}}\), as a function of diffusion rate normalized to termination rate, \(R_{m^{*}}\).