# Predictive Modeling of Light-Matter Interaction in One Dimension: A Dynamic Deep Learning Approach
## 3 Results
### Frequency Generation via Optical Wave Mixing in a Micro-Resonator
#### 3.1.1 Difference Frequency Generation Process
\[E_{2}(x=0\ \mu\mathrm{m},t=0\ \mathrm{s})=3\times 10^{8}\times sin\big{(}2\pi \big{(}1.5\times 10^{14}\big{)}t\big{)}\ \mathrm{V/m}\ \ \ \ (Infrared\ wave\ at\ 150\ \mathrm{THz})\]0 Spectrum of the output wave as measured near the cavity edge (\(x=5\) μm) at the end of the simulation duration (t = 5 ps). After a 5 ps mixing of the two high-amplitude waves, the sum and the difference frequency components are generated at 400 THz and 100 THz, respectively (along with the second harmonics of both waves at 300 THz and 500 THz). The lower-frequency wave \(E_{2}\) (150 THz) is amplified at the end of this mixing, a process known as parametric amplification. The time variation of the set of four coefficients \((a_{1},a_{2},a_{3},a_{4})\) for the first layer of the network is given in \[E_{2}(x=0\ \mu\mathrm{m},t=0\ \mathrm{s})=3\times 10^{8}\times sin\big{(}2\pi \big{(}1.5\times 10^{14}\big{)}t\big{)}\ \mathrm{V/m}\ \ \ \ (Infrared\ wave\ at\ 150\ \mathrm{THz})\]1 Notice that the coefficients are nearly constant as the change in percentage around their mean value is quite insignificant. The coefficients of the second layer are shown in \[E_{2}(x=0\ \mu\mathrm{m},t=0\ \mathrm{s})=3\times 10^{8}\times sin\big{(}2\pi \big{(}1.5\times 10^{14}\big{)}t\big{)}\ \mathrm{V/m}\ \ \ \ (Infrared\ wave\ at\ 150\ \mathrm{THz})\]2 Compared to the coefficients of the first layer, the percentage of change between the minimum and maximum values of these coefficients is much greater. However, the coefficients of the second layer tend to converge to a certain value after 5 picoseconds, which indicates that the coefficients of the first layer will continue to make the same tiny oscillations around their mean values, indicating that a reliable prediction in the long term is feasible. \[E_{2}(x=0\ \mu\mathrm{m},t=0\ \mathrm{s})=3\times 10^{8}\times sin\big{(}2\pi \big{(}1.5\times 10^{14}\big{)}t\big{)}\ \mathrm{V/m}\ \ \ \ (Infrared\ wave\ at\ 150\ \mathrm{THz})\]3 Temporal variation of the set of four coefficients in the first layer. \[E_{2}(x=0\ \mu\mathrm{m},t=0\ \mathrm{s})=3\times 10^{8}\times sin\big{(}2\pi \big{(}1.5\times 10^{14}\big{)}t\big{)}\ \mathrm{V/m}\ \ \ \ (Infrared\ wave\ at\ 150\ \mathrm{THz})\]4 Temporal variation of the set of four coefficients \((b_{1},b_{2},b_{3},b_{4})\) in the second layer. Here, the second-layer coefficients are derived from the time variation of \(a_{1}\) (\(a_{1}\to b_{1}\)), hence the notation \(\{b_{11},b_{12},b_{13},b_{14}\}\). The same coefficients \((b_{1},b_{2},b_{3},b_{4})\) can also be obtained from the time variations of \(a_{2},a_{3}\), \(a_{4}\). Finally, the coefficients of the third layer are given in \[E_{2}(x=0\ \mu\mathrm{m},t=0\ \mathrm{s})=3\times 10^{8}\times sin\big{(}2\pi \big{(}1.5\times 10^{14}\big{)}t\big{)}\ \mathrm{V/m}\ \ \ \ (Infrared\ wave\ at\ 150\ \mathrm{THz})\]5 These coefficients are used to predict whether the convergent behavior of the coefficients of the second layer will persist. There is also a tendency for convergence in this layer, which is easier to spot for the second, third, and fourth coefficients, whereas for the first coefficient, convergence is not that critical as the percentage of change is smaller. Therefore, the third-layer coefficients indicate that the convergence behavior of the second-layer coefficients will persist, which further indicates that the extremely small percentage of change for the first-layer coefficients will continue, enabling one to treat them as constant variables over time and perform a prediction for a duration that is much greater than \(\mathrm{t=5\,ps}\).