# Numerical Technique for Study of Noise Grating Dynamics in Holographic Photopolymers
## 3 Validation
These can also be solved in closed form with a symbolic math package. The resulting solutions to Equation (10) for the time evolution of the monomer harmonics, \(m_{i}(\tau)\) and the solutions to Equation (11) for the time evolution of the polymer harmonics, \(p_{i}(\tau)\), are a complete description of how the material responds to a sinusoidal intensity pattern. In the ideal recording case of perfect radical visibility, \(V_{m^{\prime}}=1\), and very fast monomer diffusion, \(R_{m}\rightarrow\infty\), the all polymer harmonics greater than two are zero. The average and first harmonic of the polymer become \(p_{0}(\tau)=p_{1}(\tau)=1-\exp\left(-\tau\right)\) and, thus, \(P(\chi,\tau)=\left[1-\exp\left(-\tau\right)\right]\left[1+\cos\left(2\;\pi\;i \chi\right)\right]\). This is the ideal recording case where the polymer records a perfectly copy of the intensity. With the monomer and polymer distributions known in space and time, the volume fraction of the matrix, \(\phi_{M}\), can be found from the assumption of incompressibility, \(\phi_{m}+\phi_{P}+\phi_{M}=1\). This completely specifies the composition of the trinary mixture. The composition and refractive indices of the constituents are then substituted into Equation (5), yielding a complicated but completely closed-form expression for material index as a function of exposure time. This lengthy equation depends on measured material parameters such as refractive indices, diffusion and reaction rates through the terms \(R_{m}\) and \(R_{m^{\prime}}\), hologram period and time. The first harmonic of this calculated index is then used to predict holographic efficiency as a function of reference beam angle via the beta value method [13]. The result is a completely closed-form prediction of hologram efficiency as a function of material parameters, writing conditions, and playback conditions. The reason for developing this fully theoretical expression for holographic efficiency is that it can be compared to the numerical code suite which ideally implements the same physics but in a completely numerical, discrete manner. Specifically, in the limit of no scattering, the code suite will reproduce the intensity pattern, chemical reactions, distribution of species and refractive index to predict Bragg diffraction efficiency via the numerical techniques described previously. A comparison of the numerical to the theoretical solutions thus serves to validate the implementation and numerical approximations of the code suite. First, we use both the theory and numerical system to compare the evolution of the index grating itself. A 10 \(\upmu\)m thick film of 20 wt% bisphenol A diacrylate in polyurethane matrix is exposed to two 405 nm Gaussian beams tilted to produce an unslanted 1 \(\upmu\)m period transmission hologram. Radical diffusion is taken to be negligible \((R_{m^{\prime}}\to 0)\) but monomer diffusivity is finite and sufficiently slow \((R_{m}=0.2)\) to generate a significant second, but limited third and higher, harmonics. The normalized exposure time, \(\tau=2.5\), is sufficiently long to consume over 90% of the monomer. The solid lines in Figure 5c show the evolution of the fundamental harmonic recorded into the polymer, \(p_{1}\), as well as the second harmonic, \(p_{2}\), caused by the finite rate of monomer diffusion. The open circles show the fully numerical result, extracted by fitting a harmonic series to a slice through the 3D sampled grid. The excellent agreement between the closed form 1 + 1D and numerical 3 + 1D approaches validates that the numerical suite has faithfully reproduced the desired physics. Next, both the theoretical and numerically derived first harmonics of the polymer and monomer are combined through the Lorentz-Lorenz equation to predict the growth of the first harmonic of the refractive index, as shown in Figure 5d. The solid line is a completely closed form prediction of grating strength and the numerical suite accurately reproduces this prediction.