## 2 Formulations
### Transfer Matrix Method
with eigenvalues \(k_{c}\) which are determined by the following eigenvalue equations, the transverse parts,  
\[\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+k_ {c}^{2}\right)\phi(x,y)_{TE}=0 \tag{7}\]  
and  
\[\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+ k_{c}^{2}\right)\phi(x,y)_{TM}=0. \tag{8}\]  
The corresponding boundary conditions for \(\phi(x,y)_{TE}\) and \(\phi(x,y)_{TM}\) are  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]0  
and  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]1  
It then follows that  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]2  
Applying the eigenvalue equations, Equations (7) and (8), and the boundary conditions, Equations (9) and (10), to the general solutions of Equations (1) and (2), the following particular solutions can be found by separation of variables,  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]3  
and  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]4  
where \(H_{0}\) and \(E_{0}\) are determined by the energy of electromagnetic waves propagating inside the waveguide, and \(m\) and \(n\) are integers. The function \(\psi(z)\) is chosen for a particular solution since it represents propagating waves in the \(z\)-direction. A transfer matrix approach is employed to discretize the dielectric function profile of the heterostructure. For an \(N\)-layer dielectric-filled waveguide, \(\varepsilon(z)\) and \(\mu(z)\) can be divided into \(p=1,2,...,(N+2)\) layers with a piecewise constant permittivity \(\varepsilon_{p}\) and constant permeability \(\mu_{p}\), respectively. The discretized one-dimensional Helmholtz's equation, for the \(p\)th region with constant permittivity \(\varepsilon_{p}\) and constant permeability \(\mu_{p}\) can be written as  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]5  
with  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]6  
where \(\psi_{p}(z)\) represents the wave function in the \(p\)th layer, and \(k_{p}\) defines the complex wavevector in the same layer along the \(z\)-direction, \(\lambda\) represents the wavelength in free space at the operating angular frequency \(\omega\), \(\varepsilon_{rp}\) and \(\mu_{rp}\) are the relative dielectric constant and permeability of the medium, respectively, and \(\lambda_{c}\) is the cutoff wavelength. The solutions of Equations (12) and (13) can be written as a superposition of the forward and backward traveling wave functions:  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]7The boundary conditions for \(\psi(z)\) at the interface between layers \(p\) and \((p+1)\) at position \(z=z_{p}\) where \(p=1,2,...,(N+1)\) are (for TE modes)  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]8  
and (for TM modes)  
\[\left\{\varepsilon(z)\vec{\
abla}\times\left[\frac{1}{\varepsilon(z)}\vec{ \
abla}\times\vec{H}\right]\right\}_{z}+\omega^{2}\varepsilon(z)\mu(z)H_{z}=0 \tag{1}\]9  
By matching the boundary conditions at each discontinuity, we arrive at  
\[\left(