## Appendix C Generalization of Stability Condition Towards Three-Dimensional FO FDTD  
In the case of three-dimensional FO FDTD, discrete Maxwell's equations are considered  
\[ \begin{split}\left(\frac{1}{\Delta t}\right)^{\alpha}\sum_{l=0}^{n }w_{l}^{\alpha}H_{x}^{n-l+\frac{1}{2}}(I,J+\frac{1}{2},K+\frac{1}{2})& =\frac{1}{\mu_{\alpha}}\left(\frac{E_{y}^{n}(I,J+\frac{1}{2},K+1)-E_{y}^{n }(I,J+\frac{1}{2},K)}{\Delta z}-\right.\\ &\left.\frac{\underline{E_{z}^{n}(I,J+1,K+\frac{1}{2})-E_{z}^{n}( I,J,K+\frac{1}{2})}}{\Delta y}\right)\\ &\left(\frac{1}{\Delta t}\right)^{\alpha}\sum_{l=0}^{n}w_{l}^{ \alpha}H_{y}^{n-l+\frac{1}{2}}(I+\frac{1}{2},J,K+\frac{1}{2})&= \frac{1}{\mu_{\alpha}}\left(\frac{E_{z}^{n}(I+1,J,K+\frac{1}{2})-E_{z}^{n}( I,J,K+\frac{1}{2})}{\Delta x}-\right.\\ &\left.\frac{E_{x}^{n}(I+\frac{1}{2},J,K+1)-E_{x}^{n}(I+\frac{1}{ 2},J,K)}{\Delta z}\right)\\ &\left(\frac{1}{\Delta t}\right)^{\alpha}\sum_{l=0}^{n}w_{l}^{ \alpha}H_{z}^{n-l+\frac{1}{2}}(I+\frac{1}{2},J+\frac{1}{2},K)&= \frac{1}{\mu_{\alpha}}\left(\frac{E_{x}^{n}(I+\frac{1}{2},J+1,K)-E_{x}^{n}(I+ \frac{1}{2},J,K)}{\Delta y}-\right.\\ &\left.\frac{E_{y}^{n}(I+1,J+\frac{1}{2},K)-E_{y}^{n}(I,J+\frac{1 }{2},K)}{\Delta x}\right)\\ &\left(\frac{1}{\Delta t}\right)^{\alpha}\sum_{l=0}^{n}w_{l}^{ \alpha}E_{x}^{n-l+1}(I+\frac{1}{2},J,K)&=\frac{1}{\varepsilon_{ \alpha}}\left(\frac{H_{z}^{n+\frac{1}{2}}(I+\frac{1}{2},J+\frac{1}{2},K)-H_{ z}^{n+\frac{1}{2}}(I+\frac{1}{2},J-\frac{1}{2},K)}{\Delta y}-\right.\\ &\left.\frac{H_{y}^{n+\frac{1}{2}}(I+\frac{1}{2},J,K+\frac{1}{2}) -H_{y}^{n+\frac{1}{2}}(I+\frac{1}{2},J+\frac{1}{2},K-\frac{1}{2})}{\Delta z} \right)\\ &\left(\frac{1}{\Delta t}\right)^{\alpha}\sum_{l=0}^{n}w_{l}^{ \alpha}E_{y}^{n-l+1}(I,J+\frac{1}{2},K)&=\frac{1}{\varepsilon_{ \alpha}}\left(\frac{H_{x}^{n+\frac{1}{2}}(I,J+\frac{1}{2},K+\frac{1}{2})-H_{ x}^{n+\frac{1}{2}}(I,J+\frac{1}{2},K-\frac{1}{2})}{\Delta z}-\right.\\ &\left.\frac{H_{z}^{n+\frac{1}{2}}(I+\frac{1}{2},[PERSON]+\frac{1}{2},K)- H_{z}^{n+\frac{1}{2}}(I-\frac{1}{2},[PERSON]+\frac{1}{2},K)}{\Delta x}\right)\\ &\left(\frac{1}{\Delta t}\right)^{\alpha}\sum_{l=0}^{n}w_{l}^{ \alpha}E_{z}^{n-l+1}(I,J,K+\frac{1}{2})&=\frac{1}{\varepsilon_{ \alpha}}\left(\frac{H_{y}^{n+\frac{1}{2}}(I+\frac{1}{2},J,K+\frac{1}{2})-H_{ y}^{n+\frac{1}{2}}(I-\frac{1}{2},J,K+\frac{1}{2})}{\Delta x}-\right.\\ &\left.\frac{H_{x}^{n+\frac{1}{2}}(I,J+\frac{1}{2},K+\frac{1}{2}) -H_{x}^{n+\frac{1}{2}}(I,J-\frac{1}{2},K+\frac{1}{2})}{\Delta y}\right)\\ \end{split}  \tag{108}\]  
where \(I\), \(J\), \(K\) denote coordinate indices in respective directions. Capital letters are used to differentiate these parameters from the imaginary unit \(j=\sqrt{-1}\). Similarly, as in the one-dimensional case, the following assumptions regarding harmonic solutions are taken  
\[ \begin{split} H_{x}^{n}(I,J,K)&=H_{x0}\varrho^{i( \omega n\Delta t-\vec{l}_{x}IAx-\vec{j}_{y}I\Delta y-\vec{k}_{z}K\Delta z)}\\ H_{y}^{n}(I,J,K)&=H_{y0}\varrho^{i(\omega n\Delta t- \vec{l}_{x}IAx-\vec{j}_{y}I\Delta y-\vec{k}_{z}K\Delta z)}\\ H_{z}^{n}(I,J,K)&=H_{z0}\varrho^{i(\omega n\Delta t- \vec{l}_{x}IAx-\vec{j}_{y}I\Delta y-\vec{k}_{z}K\Delta z)}\\ E_{x}^{n}(I,J,K)&=E_{x0}\varrho^{i(\omega n\Delta t- \vec{l}_{x}IAx-\vec{j}_{y}I\Delta y-\vec{k}_{z}K\Delta z)}\\ E_{y}^{n}(I,J,K)&=E_{y0}\varrho^{i(\omega n\Delta t- \vec{l}_{x}IAx-\vec{j}_{y}I\Delta y-\vec{k}_{z}K\Delta z)}\\ E_{z}^{n}(I,J,K)&=E_{z0}\varrho^{i(\omega n\Delta t- \vec{l}_{x}IAx-\vec{j}_{y}I\Delta y-\vec{k}_{z}K\Delta z)}\end{split}  \tag{109}\]  
\[\delta_{y} = (\Delta t)^{\alpha}\frac{e^{-j[\frac{\gamma}{y}\frac{1}{2}\Delta y]}-e ^{j[\frac{\gamma}{y}\frac{1}{2}\Delta y]}}{z_{s}\Delta y} \tag{100}\] \[\delta_{z} = (\Delta t)^{\alpha}\frac{e^{-j[\frac{\gamma}{y}\frac{1}{2}\Delta z ]}-e^{j[\frac{\gamma}{y}\frac{1}{2}\Delta z]}}{z_{s}\Delta z}. \tag{101}\]  
The system of Equations (101) is homogeneous. In order to obtain any nontrivial (nonzero) solution, the determinant of the matrix \(\mathbf{U}\) has to be equal 0. The matrix \(\mathbf{U}\) can be interpreted as the block matrix  
\[\mathbf{U}=