In any propagation wave functions, the propagation factor can also be set as on the z direction. Thus we assume the electric field is given,
\begin{equation} \mathbf{E}=E(x,y)\exp{i \beta z} \end{equation}
where $A(x,y)$ is field distribution on the propagation plane and $\exp{i \beta z}$ is propagation factor.
Hence, all the components of EMF oscillate on the z direction at a wave vector $\beta$. Then simple equations of EMF components can be derived from the Maxwell Equations in the homogeneous media.
\(\begin{eqnarray} E_x &= \frac{i}{k_c^2}(\beta \frac{\partial E_z}{\partial x} + \omega \mu \frac{\partial H_z}{\partial y}) \\ E_y &= \frac{i}{k_c^2}(\beta \frac{\partial E_z}{\partial y} - \omega \mu \frac{\partial H_z}{\partial y}) \\ H_x &= \frac{i}{k_c^2}(\beta \frac{\partial H_z}{\partial x} - \omega \mu \frac{\partial E_z}{\partial y}) \\ H_y &= \frac{i}{k_c^2}(\beta \frac{\partial H_z}{\partial y} + \omega \mu \frac{\partial E_z}{\partial x}) \end{eqnarray}\) thus,
where $ k_c^2 = n^2 k^2 - \beta ^2 $, which is defined as cut-off wave vector.
Here we can see, there are two independent solutions to the above equation, where $E_z$ and $H_z$ are two independent variables. In result, these two modes are named TE and TM modes. Separately, only $E_z$ or $H_z$ doesn’t equal to zero.
In the no-charged and non-current surface, the boundary condition on two different media can be listed as
\[\begin{eqnarray*} \mathbf{n_{21}} \cdotp (\mathbf{D_2} - \mathbf{D_1}) &= 0 \\ \mathbf{n_{21}} \times (\mathbf{E_2} - \mathbf{E_1}) &= 0 \\ \mathbf{n_{21}} \cdotp (\mathbf{B_2} - \mathbf{B_1}) &= 0 \\ \mathbf{n_{21}} \times (\mathbf{H_2} - \mathbf{H_1}) &= 0 \end{eqnarray*}\]Set the normal direction of the surface as the reference,
rewrite,
where the $\perp, \parallel$ mean the normal and tangential components of EMF.
e.x, if we study the xz surface (y direction is the normal vector) of the TE mode, for $E_z = 0$,
\[\begin{eqnarray} \varepsilon_1 E_{1,y}=\varepsilon_2 E_{2,y} , E_{1,x}=E_{2,x} , E_{1,z}=E_{2,z} =0 \\ \mu_1 H_{1,y}=\mu_2 H_{2,y} , H_{1,x}=H_{2,x} , H_{1,z}=H_{2,z} \neq 0 \end{eqnarray}\]Furthermore, the solution of either TE or TE modes can be transfer as
\begin{equation} \nabla^2_{x,y} ~A + n^2 k^2 - \beta^2 A = 0 \end{equation}
and where n is the refractive index of material and $\beta$ is the propagation constant.
By using Marcatili model, assuming the refractive index of four corners are the same, $n_c$, see. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969), the rectangular waveguides can be calculated like two planar waveguides. Approximation includes two simplifications,
e.x, the EMF of TE modes can be written, \(\begin{eqnarray} \mathbf{E} =(E_x, E_y, 0) \\ \mathbf{H} =(H_x, H_y, H_z) \\ \mathbf{k} =(k_x, k_y, k_z) \end{eqnarray}\) divided as the superposition of two EMF, like
\[\begin{eqnarray} \mathbf{E_1}=(E_x, 0, 0) \mathbf{E_2}=(0, E_y, 0) \\ \mathbf{H_1}=(0, H_{y}, H_{1z}) \mathbf{H_2}=(H_{x}, 0, H_{2z}) \\ \mathbf{k_1}=(0, k_{y}, k_{1z}) \mathbf{k_2}=(k_{x}, 0, k_{2z}) \end{eqnarray}\]which can satisfy the transverse field condition and energy conservation simultaneously. Hence, the solution turns into solving the planar waveguide field distribution.
In a passive field, like silicon waveguide, there is nothing more complex than the Helmholtz equation in the vector field, where the eigen-solution only depends on the boundary condition.
From the Maxwell equations, all the components can be represented by z-component. Finally, the problem ends up into only one Helmholtz equation with two groups of boundary conditions, on xz surface and yz surface.
to be added
\begin{equation} 2pd=m\pi + \tan^{-1}(\kappa_1/p) + \tan^{-1}(\kappa_2/p) \end{equation}
$d$ is half of the thickness and $p$ is the
The latter two terms are phase shift on the boundary.
For symmetric waveguides
\begin{equation} \kappa = p \tan(pd - m\pi/2) \end{equation}
Define numerical $NA=\sqrt{n_{co}^2-n_{cld}^2}$,
$v=d\sqrt{p^2+\kappa^2}=k_0 d \sqrt{n_{co}^2-n_{cld}^2}$ is constant and $u = pd$ ,
\begin{equation} \sqrt{v^2-u^2}=u\tan(u-m\pi/2) \end{equation}
\begin{equation} 2pd=m\pi + \tan^{-1}(\frac{n_c^2\kappa_1}{n_1^2p}) + \tan^{-1}(\frac{n_c^2 \kappa_2}{n_2^2p}) \end{equation}
The latter two terms are phase shift on the boundary.
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