FDFD method for waveguide mode analysis

Introduction of the program

This program simulates the mode of waveguides and the material could be anisotropic:

$$ \varepsilon = \left[\begin{matrix} &\varepsilon_{xx} & 0 & 0 \\ & 0 & \varepsilon_{yy} &0 \\ & 0 & 0 &\varepsilon_{zz} \end{matrix} \right] $$

Theoretical formulation

We begin from the Maxwell equations

$$ \nabla \times H =i \times \omega [ \varepsilon]E $$

$$ \nabla \times E=-j \omega \mu H $$

$$ \nabla \cdot B = 0 $$

$$ \nabla \cdot [\varepsilon]E =0 $$

for waveguide modes, we set $H(x,y,z)=H(x,y)e^{-i\beta z}$， and we can obtain the equations of $H_{x},H_{y}$

$$ \frac{\partial^2 h_{x}}{\partial x^2}+\frac{\varepsilon_y}{\varepsilon_z} \frac{\partial^2 h_{x}}{\partial y^2}+(1-\frac{\varepsilon_y}{\varepsilon_z})\frac{\partial^2 h_{y}}{\partial x \partial y}+k^2 \varepsilon_{x} h_{x}^{2}=\beta^{2}h_{x} $$

$$ \frac{\partial^2 h_{y}}{\partial y^2}+\frac{\varepsilon_x}{\varepsilon_z} \frac{\partial^2 h_{y}}{\partial x^2}+(1-\frac{\varepsilon_x}{\varepsilon_z})\frac{\partial^2 h_{x}}{\partial x \partial y}+k^2 \varepsilon_{y} h_{y}^{2}=\beta^{2}h_{y} $$

after derivations, the final five-point equations to be used in program should be

$$ H_{xW}+H_{xE}+\frac{d_{x}}{d_{y}}\left( \frac{\frac{1}{\epsilon_{1z}}}{\frac{1}{\epsilon_{1y}+\frac{1}{\epsilon{2y}}}} +\frac{\frac{1}{\epsilon_{4z}}}{\frac{1}{\epsilon_{4y}+\frac{1}{\epsilon{3y}}}}\right)H_{xN}+ \frac{d_{x}}{d_{y}}\left( \frac{\frac{1}{\epsilon_{2z}}}{\frac{1}{\epsilon_{1y}+\frac{1}{\epsilon{2y}}}}+ \frac{\frac{1}{\epsilon_{3z}}}{\frac{1}{\epsilon_{4y}+\frac{1}{\epsilon{3y}}}} \right)H_{xS}+ $$

$$ \left( \frac{k^2}{\frac{1}{\epsilon_{4y}}+\frac{1}{\epsilon{3y}}}(d_{x})^2 -2+\frac{k^2}{\frac{1}{\epsilon_{1y}}+\frac{1}{\epsilon_{2y}}}(d_{x})^2- \frac{\frac{1}{\epsilon_{3z}}+\frac{1}{\epsilon_{4z}}}{\frac{1}{\epsilon_{4y}}+\frac{1}{\epsilon_{3y}}} \left(\frac{{d_{x}}^2}{{d_{y}}^2}\right)- \frac{\frac{1}{\epsilon_{2z}}+\frac{1}{\epsilon_{1z}}}{\frac{1}{\epsilon_{2y}}+\frac{1}{\epsilon_{1y}}} \left(\frac{{d_{x}}^2}{{d_{y}}^2}\right) \right) H_{xP} $$

$$ \frac{d_{x}}{2d_{y}}\left(\frac{\frac{1}{\epsilon_{4z}}-\frac{1}{\epsilon_{3z}}}{\frac{1}{\epsilon_{4y}}+\frac{1}{\epsilon_{3y}}}-\frac{\frac{1}{\epsilon_{2y}}-\frac{1}{\epsilon_{1y}}}{\frac{1}{\epsilon_{1y}}+\frac{1}{\epsilon_{2y}}}\right)H_{yW}+ \frac{d_{x}}{2d_{y}}\left(\frac{\frac{1}{\epsilon_{2z}}-\frac{1}{\epsilon_{1z}}}{\frac{1}{\epsilon_{1y}}+\frac{1}{\epsilon_{2y}}}-\frac{\frac{1}{\epsilon_{4y}}-\frac{1}{\epsilon_{3y}}}{\frac{1}{\epsilon_{3y}}+\frac{1}{\epsilon_{4y}}}\right)H_{yE}=\beta^2(d_{x})^2H_{xP} $$

$$ H_{yN}+H_{yS}+\frac{d_{y}}{d_{x}}\left( \frac{\frac{1}{\epsilon_{1z}}}{\frac{1}{\epsilon_{1x}+\frac{1}{\epsilon{4x}}}} +\frac{\frac{1}{\epsilon_{2z}}}{\frac{1}{\epsilon_{2x}+\frac{1}{\epsilon{3x}}}}\right)H_{yE}+ \frac{d_{y}}{d_{x}}\left( \frac{\frac{1}{\epsilon_{4z}}}{\frac{1}{\epsilon_{1x}+\frac{1}{\epsilon{4x}}}}+ \frac{\frac{1}{\epsilon_{3z}}}{\frac{1}{\epsilon_{2x}+\frac{1}{\epsilon{3x}}}} \right)H_{yW}+ $$

$$ \left( \frac{k^2}{\frac{1}{\epsilon_{4x}}+\frac{1}{\epsilon{1x}}}(d_{y})^2 -2+\frac{k^2}{\frac{1}{\epsilon_{3x}}+\frac{1}{\epsilon_{2x}}}(d_{y})^2- \frac{\frac{1}{\epsilon_{1z}}+\frac{1}{\epsilon_{4z}}}{\frac{1}{\epsilon_{4x}}+\frac{1}{\epsilon_{1x}}} \left(\frac{{d_{y}}^2}{{d_{x}}^2}\right)- \frac{\frac{1}{\epsilon_{2z}}+\frac{1}{\epsilon_{3z}}}{\frac{1}{\epsilon_{2x}}+\frac{1}{\epsilon_{3x}}} \left(\frac{{d_{y}}^2}{{d_{x}}^2}\right) \right) H_{yP} $$

$$ \frac{d_{x}}{2d_{y}}\left(\frac{\frac{1}{\epsilon_{2z}}-\frac{1}{\epsilon_{3z}}}{\frac{1}{\epsilon_{2x}}+\frac{1}{\epsilon_{3x}}}-\frac{\frac{1}{\epsilon_{4x}}-\frac{1}{\epsilon_{1x}}}{\frac{1}{\epsilon_{1x}}+\frac{1}{\epsilon_{4x}}}\right)H_{xN}+ \frac{d_{x}}{2d_{y}}\left(\frac{\frac{1}{\epsilon_{4z}}-\frac{1}{\epsilon_{1z}}}{\frac{1}{\epsilon_{1x}}+\frac{1}{\epsilon_{4x}}}-\frac{\frac{1}{\epsilon_{2x}}-\frac{1}{\epsilon_{3x}}}{\frac{1}{\epsilon_{3x}}+\frac{1}{\epsilon_{2x}}}\right)H_{xS}=\beta^2(d_{y})^2H_{yP} $$

and the position of $H_{x/y,N/S/W/E}$ in the grid point is as follows

the final eigenvalue equations could be written as $$ \textbf{M} \left [\begin{matrix}H_{x}\ H_{y}\end{matrix}\right]=\beta^2 \left [\begin{matrix}H_{x}\ H_{y}\end{matrix}\right] $$

Simple example

Rectangle waveguide on chip

The program WaveguideSet_RectGuide defines general on-chip waveguide structures and the waveguide modes can be simulated

Step index fiber

The program WaveguideSet_StepIndexFiber defines general step index fiber structures and the waveguide modes can be simulated