# Numerical Technique for Study of Noise Grating Dynamics in Holographic Photopolymers
###### Abstract  
Although the angular distribution of noise gratings in holographic photopolymer is understood to arise from Bragg matching, the details of scatter strength and dynamics are not fully understood. This confounds development of materials and recording techniques that minimize haze. Here, the kinetics are studied using a multi-physics numerical approach coupling diffraction of light from the dynamic material including scatter centers, reactions of chemical species initiated by this light, diffusion and swelling of these constituents, and the formation of the refractive index from the resulting composition. The approach is validated in the case of two-beam transmission holography by comparison to traditional harmonic series and rigorous coupled-mode approaches. Two beam holography in the presence of scatter is then used to study haze development. This reveals that haze due to weak noise gratings grows significantly above initial scatter only in reaction-limited materials, consistent with proposed Bragg-matched amplification mechanisms. Amplified haze is found to be proportional to initial scatter, quantifying the impact of clean sample fabrication. Conversely, haze is found to grow super-linearly with sample thickness, illustrating the significant challenge for applications requiring low haze in large thickness. h +
Footnote †: journal:  
h +  
The observed scatter patterns can be explained by the Ewald sphere analysis shown in Figure 1. In the left two images, object and reference beams, shown as blue arrows, write a holographic grating with positive and negative side bands, shown as the two heavy black arrows. Scatter in a cone around the reference beam (top) or object beam (bottom), shown as a circular patch on the Ewald sphere, interferes with the object beam to write weak noise gratings. The positive and negative side bands of these noise gratings, shown as thin black lines, reside on two-dimensional spherical surfaces centered on the holographic grating K vector. The interference of the object beam with the reference (top) and object (bottom) scatter thus generates four two-dimensional noise grating manifolds, shown as shaded spherical patches. The right-hand plots show that a curved one-dimensional slice of two of these manifolds is Bragg matched when replayed by the reference beam. The slice, shown as heavy red lines, is found at the intersection of the Ewald sphere and the diffracted k vectors that occupy the curved spherical manifolds. Within these arcs, Bragg-matched diffraction of the reference beam off of the noise grating manifolds increases the magnitude of the noise. This larger scattered intensity writes new and stronger noise holograms and the process repeats, amplifying the scatter in a distinct pattern. The figure shows only half of the possible noise grating manifolds for clarity. Interference of the reference beam with scatter follows the same process, mirrored around the \(xz\) plane. While this analysis and the previous studies explain the angular distribution of recorded scatter patterns, they do little to guide materials and process development that would maximize SNR. The amplitude of the noise gratings is expected to be strongly dependent on the statics of the scatter centers, the dynamics of the recording process, Bragg selectivity of the noise gratings, and recording parameters such as period and intensity. These are difficult to study experimentally since it is challenging to know or precisely control the details of the initial weak scattering. Additionally, experimentally exploring the parameter space of materials formulations and processing conditions is costly when it is not clear what variables should be probed and in what ranges they would be expected to impact the SNR. The goal of this paper is to therefore to establish a numerical approach to study these questions.