Basic Equations
For the wave dynamics of the electric and magnetic fields in a dielectric waveguide, one seeks solutions of Maxwell’s equations with harmonic time dependence, so-called optical modes.
Maxwell’s equation in a dielectric medium are given by
$$
\begin{align}
\nabla \cdot \vec{D} &= \rho_f \tag{1.a} \\
\nabla \cdot \vec{B} &= 0 \tag{1.b} \\
\nabla \times \vec{E} &= - \partial_t \vec{B} \tag{1.c} \\
\nabla \times \vec{H} &= \vec{j}_f + \partial_t \vec{D} \tag{1.d}
\end{align}
$$
Only linear isotropic media are considered here, which yields constitutive relation:
$$
\begin{align}
\vec{D} &= \varepsilon_0 \varepsilon_r \vec{E} \tag{2.a} \\
\vec{B} &= \mu_0 \mu_r \vec{H} \tag{2.b}
\end{align}
$$
where $(\varepsilon_r) \varepsilon_0$ is the (relative) permittivity and $(\mu_r)\mu_0$ is the (relative) permeability.
They are related to refractive index $n=\sqrt{\varepsilon_r \mu_r}$ and vacuum speed of light $c= 1/\sqrt{\varepsilon_0 \mu_0}$
Furthermore it is assumed that no external currents and charges are present, $\vec{j}_f=0$ and $\rho_f=0$.
The boundary conditions at the interface between two isotropic materials with dielectric constants $\varepsilon_1$ and $\varepsilon_2$ are
$$
\begin{align}
\vec{\nu} \cdot (\varepsilon_1 \vec{E_1} - \varepsilon_2 \vec{E_2}) = 0 \tag{3.a} \\
\vec{\nu} \cdot (\vec{H_1} - \vec{H_2}) = 0 \tag{3.b} \\
\vec{\nu} \times (\vec{E_1} - \vec{E_2}) = 0 \tag{3.c} \\
\vec{\nu} \times (\vec{H_1} - \vec{H_2}) = 0 \tag{3.d}
\end{align}
$$
where $\varepsilon=\varepsilon_0 n^2$, assuming $\mu_r = 1$, which is typically the case.
Derivation of Wave Equation
$$
\begin{align}
\nabla \times \nabla \times \vec{E} &= \nabla (\nabla \cdot \vec{E}) - \nabla^2 \vec{E} = -\nabla^2 \vec{E} \tag{4.a} \\
\nabla \times \nabla \times \vec{H} &= \nabla (\nabla \cdot \vec{B})/\mu_0 - \nabla^2 \vec{H} = -\nabla^2 \vec{H} \tag{4.b}
\end{align}
$$
On the other hand,
$$
\begin{align}
\nabla \times \nabla \times \vec{E} &= \nabla \times (-\partial_t \vec{B}) = -\mu_0 \partial_t (\nabla \times \vec{H}) = -\mu_0 \partial^2_t \vec{D} = - \frac{n^2}{c^2} \partial^2_t \vec{E} \tag{5.a} \\
\nabla \times \nabla \times \vec{H} &= \nabla \times (\partial_t \vec{D}) = \varepsilon \partial_t (\nabla \times \vec{E}) = -\varepsilon \partial^2_t \vec{B} = - \frac{n^2}{c^2} \partial^2_t \vec{H} \tag{5.b}
\end{align}
$$
Eqs. (3.4a)-(3.5b) can be compactly written as the wave equation
$$
\begin{align}
\nabla^2 \vec{\Psi} \, (\vec{r},t) - \frac{n^2(\vec{r})}{c^2}\partial^2_t \vec{\Psi} \, (\vec{r},t) = 0 \tag{6}
\end{align}
$$
For $\vec{\Psi} \in$ {$\vec{E}, \vec{H}$}. A separation of variables leads to $\vec{\Psi}\,(\vec{r},t)=\vec{\Psi}\,(\vec{r}) exp(-i\omega t)$
The spatial distribution, the actual mode $\vec{\Psi}\,(\vec{r})$, is determined by the equaton
$$
\begin{align}
\nabla^2 \psi + n^2 k^2 \psi = 0 \tag{7}
\end{align}
$$
for every component $\psi$ of the field vector $\vec{\Psi}$
An eigenmode of waveguide structure is a propagating or evanscent wave of which the transversal shape does not change during propagation.
An eigenmode propagating in z-direction is represented by
$$
\begin{align*}
\vec{E} (\vec{r}) &= \vec{E}(x,y) e^{-i\beta z} \\
\vec{H} (\vec{r}) &= \vec{H}(x,y) e^{-i\beta z}
\end{align*}\tag{8}
$$
Which makes equation (7)
$$
\begin{align}
\nabla^2 \psi (x,y) + (n^2 k^2 - \beta^2) \psi (x,y) = 0 \tag{9}
\end{align}
$$
Then the effective refractive index ($n_{eff} = \frac{\beta}{k}$) are in range
$$
\begin{align*}
n_{core} > n_{eff} > n_{clad} \quad (\text{for guided modes}) \\
\\
n_{clad} > n_{eff} \quad (\text{for radiative modes})
\end{align*}
$$
Guided modes and radiative modes form a complete set of functions.
$$
\begin{align*}
\vec{E}(x,y,z)=\sum_{m} a_m \vec{E}_m(x,y)e^{-i\beta_m z} + \int a(k) \vec{E}_k(x,y)e^{-ikz} \, dk \tag{10}
\end{align*}
$$
The Slab Waveguide
