## 2 Materials and Methods  
### Ring Resonator  
[PERSON]'s paper in 1969 introduces the simulation of integrated ring resonators for bandpass filters. Figure 1 illustrates the layout of the channel dropping filter proposed by [PERSON], which has since become the standard configuration for such filters [44]. This configuration consists of two straight waveguides, also known as bus or port waveguides, coupled to the ring resonator through either directional coupler utilizing the evanescent field or multimode interference (MMI) couplers. Figure 2 depicts the basic configuration of a ring resonator channel dropping filter, as described by [PERSON] [45; 46]. In this configuration, there is unidirectional coupling between a ring resonator with a radius r and a waveguide. The interaction between the waveguide and the ring resonator can be described using a matrix relation as given in Equation (1):  
\[\left( \begin{array}{c}E_{t1}\\ E_{t2}\end{array} \right)=\left( \begin{array}{cc}t&\kappa\\ -\kappa^{*}&t^{*}\end{array} \right)\left( \begin{array}{c}E_{t1}\\ E_{t2}\end{array} \right) \tag{1}\]  
The complex mode amplitudes \(E\) in the system are normalized, meaning that their squared magnitudes represent the modal power. The values of the coupler parameters \(t\) and \(\kappa\) are specific to the coupling mechanism employed. The asterisk (*) signifies the conjugated complex value of \(t\) and \(\kappa\). The matrix exhibits symmetry because the considered networks adhere to the principle of reciprocity. As a result, the matrix elements are symmetric based on Equation (2):  
\[\left|\kappa^{2}\right|+\left|t^{2}\right|=1 \tag{2}\]  
To simplify the model further, the value of \(E_{t1}\) is set to 1. This choice simplifies the analysis by fixing the initial mode amplitude at the input port of the ring resonator. With this assumption, the round trip in the ring can be expressed by Equation (3), which describes the behavior of the signals propagating through the ring:  
\[E_{t2}=\alpha\cdot e^{\theta}E_{t2} \tag{3}\]  
Here, \(\alpha\) is the loss coefficient of the ring, and it represents the amount of signal attenuation or power loss within the ring. The variable \(\alpha\) is 1 when there is no loss. The variable \(\theta\) is defined as the product of the angular frequency \(\omega\), the circumference of the ring \(L\), and the reciprocal of the phase velocity \(c\). Specifically, \(\theta=\omega\)L/\(c\). The phase velocity of the ring mode, denoted as \(c\), is related to the speed of light in a vacuum, denoted as \(c_{0}\), and  
Figure 1: Basic ring resonator structure in drop channel filtering [45]. Figure 2: General schematic of ring resonator coupling [46]. the effective refractive index \(n_{\mathit{eff}}\). More precisely, \(c=c_{0}/n_{\mathit{eff}}\). By incorporating the effective refractive index, \(n_{\mathit{eff}}\), into the coupling relations of the ring, the characteristics of the ring resonator can be described more accurately. To relate the vacuum wavenumber \(k\) to the wavelength \(\lambda\), the following relationship is employed: \(k=2\pi/\lambda\). The vacuum wavenumber represents the spatial frequency of the electromagnetic wave. By incorporating these definitions and relationships, the effective refractive index, \(n_{\mathit{eff}}\), can be introduced into the coupling relations of the ring resonator, allowing for a more comprehensive description of the ring's behavior as shown in Equation (4):  
\[\beta=k\cdot n_{\mathit{eff}}=\frac{2\pi\cdot n_{\mathit{eff}}}{\lambda} \tag{4}\]  
Various types of ring resonators are proposed in the literature to meet the requirements for different application areas [44]. An add-drop ring resonator, which is another configuration of the ring resonator, is made up of a single input, single output waveguide, and one ring resonator structure as represented in Figure 3.  
The add-drop ring resonator plays an important role in antenna design due to its ability to provide frequency reconfigurability, a highly desirable feature in wireless communication systems.