## 3 Wave Propagation in Non-Linear Dispersive Media
### FDTD Equations:  
\[\frac{E_{2,k}(i+1,j)-2E_{2,k}(i,j)+E_{2,k}(i-1,j)}{\Delta x^{2}}-\mu_{0}e_{ \infty}(i,j)\frac{E_{2,k}(i,j+1)-2E_{2,k}(i,j)+E_{2,k}(i,j-1)}{\Delta t^{2}}= \tag{18a}\] \[\mu_{0}\sigma(i,j)\frac{E_{2,k}(i,j)-E_{2,k}(i,j-1)}{\Delta t}+\mu_{0}\frac{ P_{2,k}(i,j+1)-2P_{2,k}(i,j)+P_{2,k}(i,j-1)}{\Delta t^{2}}=\] (18b) \[\frac{P_{2,k}(i,j+1)-2P_{2,k}(i,j)+P_{2,k}(i,j-1)}{\Delta t^{2}}+\gamma\frac{ P_{1,k}(i,j)-P_{2,k}(i,j-1)}{\Delta t}+4\pi^{2}f_{0}^{2}\Big{(}P_{2,k}(i,j)\Big{)}-\] \[\frac{4\pi^{2}f_{0}^{2}}{N\Delta t}\Big{(}P_{2,k}(i,j)\Big{)}^{2} -\frac{4\pi^{2}f_{0}^{2}}{N\Delta^{2}e^{2}\Delta t}\Big{(}P_{2,k}(i,j)\Big{)}^ {3}=\frac{N\Delta t}{2}\Big{(}E_{2,k}(i,j)\Big{)}.\] \[\frac{E_{1,k}(i+1,j)-2E_{1,k}(i,j)+E_{1,k}(i-1,j)}{\Delta x^{2}} -\mu_{0}e_{\infty}(i,j)\frac{E_{1,k}(i,j+1)-2E_{1,k}(i,j)+E_{1,k}(i,j-1)}{ \Delta t^{2}}=\] (18c) \[\mu_{0}\sigma(i,j)\frac{E_{1,k}(i,j)-E_{1,k}(i,j-1)}{\Delta t}+\mu_{0} \frac{P_{1,k}(i,j+1)-2P_{1,k}(i,j)+P_{1,k}(i,j-1)}{\Delta t^{2}}=\] (18d) \[\frac{P_{1,k}(i,j+1)-2P_{1,k}(i,j)+P_{1,k}(i,j-1)}{\Delta t^{2}}+ \gamma\frac{P_{1,k}(i,j)-P_{1,k}(i,j-1)}{\Delta t}+4\pi^{2}f_{0}^{2}\Big{(}P_{ 1,k}(i,j)\Big{)}\] \[-\frac{4\pi^{2}f_{0}^{2}}{N\Delta t}\Big{(}P_{1,k}(i,j)\Big{)}^{2} +2P_{1,k}(i,j)P_{2,k}(i,j)\Big{)}-\frac{4\pi^{2}f_{0}^{2}}{N^{2}e^{2}\Delta t} \Big{(}\Big{(}P_{1,k}(i,j)\Big{)}^{3}\] \[+3\Big{(}P_{1,k}(i,j)\Big{)}^{2}P_{2,k}(i,j)+3P_{1,k}(i,j)\Big{(} P_{2,k}(i,j)\Big{)}^{2}\Big{)}=\frac{N\Delta t^{2}}{m}\Big{(}E_{1,k}(i,j)\Big{)}.\]  
\(x\): spatial coordinate, \(t\): time, \(k\): iteration number, \(E_{k}(x,t)=E_{k}(i\Delta x,j\Delta t)\to E_{k}(i,j)\)\(E_{2,k}:\)_High intensity pump wave at iteration \(k\), \(E_{1,k}:\) Stimulus wave at iteration \(k\)_  
**Constraints**  
\[b_{1}(\mathbf{\
u})=a_{1}g_{1}(\mathbf{\
u})\leq c_{1}\]  
\[b_{2}(\mathbf{\
u})=a_{2}\ \ g_{2}(\mathbf{\
u})\leq c_{2}\]  
\[b_{N}(\mathbf{\
u})=a_{N}g_{N}(\mathbf{\
u})\leq c_{N}\]  
**Cost function**: \(\big{|}E_{1}(\mathbf{\
u})\big{|}\)  
Figure 4: Flowchart diagram of Broyden–Fletcher-Goldfarb–Shanno (BFGS)-based non-linear programming integrated in finite-difference time-domain (FDTD) analysis.  
**Augmented cost function**: (penalty function method)  
\[f=\big{|}E_{1}(\mathbf{\
u})\big{|}+L_{1}\!\left\{\sum_{i=1}^{N}\delta_{i}(g_{i}(\bm {v})-c_{i})^{2}\right\}+L_{2}\!\left\{\sum_{j=1}^{M}\zeta_{j}\Big{(}b_{j}(\mathbf{ \
u})-a_{j}\Big{)}^{2}\right\}\]  
\[\delta_{i}=\left\{\begin{array}{ccc}0&if&g_{i}(\mathbf{v})\leq c_{i}\\ >0&if&g_{i}(\mathbf{v})>c_{i}\end{array}\right\}\]  
**Iterations**:  
\[\mathbf{v}_{k+1}=\mathbf{v}_{k}+\alpha_{k}\mathbf{p}_{k}\]  
\[\mathbf{p}_{k}=-H_{k}\
abla f_{k}\]  
\[\mathbf{s_{k}}=\mathbf{x_{k+1}}\mathbf{-x_{k}}\]  
\[\mathbf{y_{k}}=\
abla f_{k+1}\mathbf{-\
abla f_{k}}\]  
\[H_{k+1}=\Big{(}\mathbf{I}-\rho_{k}\mathbf{s_{k}}\mathbf{y_{k}}^{\mathbf{T}}\Big{)}H_{k}\big{(} \mathbf{I}-\rho_{k}\mathbf{y_{k}}\mathbf{s_{k}}^{\mathbf{T}}\Big{)}+\rho_{k}\mathbf{s_{k}}\mathbf{s_{k }}^{\mathbf{T}}\ (\text{BFGS})\]  
\[\rho_{k}=\frac{1}{\mathbf{y_{k}}^{\mathbf{T}}\mathbf{s_{k}}}\]  
Wolfe conditions for \(\alpha_{k}\): \(f(x_{k}+\ \alpha_{k}\,p_{k})\leq f(x_{k})+c_{1}\ \alpha_{k}\
abla f_{k}^{\mathbf{T}}p_{k}\)  
\[\Big{|}\
abla f(x_{k}+\ \alpha_{k}\,p_{k})^{\mathbf{T}}p_{k}\Big{|}\leq c_{2}\big{|} \
abla f_{k}^{\mathbf{T}}p_{k}\big{|}\ 0<c_{1}<c_{2}<1\]