# Modeling of advanced photonic integrated resonators using scattering matrix methods

David Moss

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David Moss. Modeling of advanced photonic integrated resonators using scattering matrix methods. 2025. hal-04864401

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# Modeling of advanced photonic integrated resonators using scattering matrix methods

David J. Moss

Optical Sciences Centre, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia;
Email: dmoss@swin.edu.au

**Abstract:** We propose a universal approach for modeling complex integrated photonic resonators based on the scattering matrix method. By dividing devices into basic elements including directional couplers and connecting waveguides, our approach can be used to model integrated photonic resonators with both unidirectional and bidirectional light propagation, with the simulated spectral response showing good agreement with experimental results. A simplified form of our approach, which divides devices into several independent submodules such as microring resonators and Sagnac interferometers, is also introduced to streamline the calculation of spectral transfer functions. Finally, we discuss the deviations introduced by approximations in our modeling, along with strategies for improving modeling accuracy. Our approach is universal across different integrated platforms, providing a useful tool for designing and optimizing integrated photonic devices with complex configurations.

**Keywords:** integrated photonics; scattering matrix method; microring resonators

## 1. Introduction

With compact footprint and versatile configuration, integrated photonic resonators have become critical building blocks for photonic integrated circuits (PICs), with applications in a wide range of fields such as optical communications [1-3], photonic computing [4-6], nonlinear optics [7-9], sensing [10-12], and optical neural networks [13-15].

Modeling the spectral response of integrated photonic resonators is of fundamental importance for their use in different applications. The scattering matrix method (SMM) (also known as the transfer matrix method) [16, 17], which is derived based on the Maxwell's equations for electromagnetic waves, has been widely employed to model the spectral response of integrated photonic resonators with different device configurations [17-22].

Although in principle the SMM can be applied to model integrated photonic devices with arbitrary planar configurations [16, 23], previous studies mainly focused on modeling devices with simple configurations. A key limitation comes from the fact that the traditional SMM relies on the manual derivation of scattering matrices, and the obtained scattering matrices need to be multiplied in a specific sequence [24]. For devices with simple configurations, this allows for relatively straightforward calculation of spectral transfer functions. However, for complex integrated photonic resonators, particularly those with bidirectional light propagation, this process becomes much more complicated, which greatly limits its broader applicability.

In this paper, a universal approach based on the SMM is proposed to model integrated photonic resonators with complex structures. The modeling is achieved by dividing a device configuration into basic elements including directional couplers and connecting waveguides, followed by solving a system of linear scattering matrix equations using computational tools. Our approach can be applied to model devices with both
unidirectional and bidirectional light propagation, and the simulated spectral response aligns well with measured results for practical devices. The modeling in our approach can also be simplified by dividing the device configuration into independent submodules such as microring resonators (MRRs) and Sagnac interferometers (SIs). Finally, we discuss the limitations of our approach induced by approximations in the modeling, together with strategies for improving modeling accuracy. Our approach provides an effective way for designing and optimizing complex integrated photonic devices across various integrated platforms.

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## 2. Modeling of devices with unidirectional light propagation

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In this section, we use the device configuration shown in Figure 1(a) as an example to show how to model a complex integrated photonic resonator based on the SMM in four steps. The device consists of a microring resonator (MRR) nested within another, with light propagating in only one direction in each waveguide.

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First, we divide the device into several directional couplers and connecting waveguides, which are the basic elements in photonic integrated circuits that form integrated photonic resonators. The optical fields at the dividing points between these elements are denoted as $E_i (i = 1–16)$.

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<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a) Schematic illustration of an integrated photonic resonator consisting of two microring resonators (MRRs) with an input from Port 1. The device is divided into several directional couplers and connecting waveguides, with $E_i (i = 1–16)$ denoting the optical fields at the dividing points. (b) – (c) zoom-in views for a directional coupler and a connecting waveguide, respectively.</figcaption></figure>

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Next, we establish a set of scattering matrix equations for the basic elements obtained in the first step, as shown in **Table 1**. These scattering matrix equations describe the relation between $E_i$ at the input / output ports of the directional couplers and the connecting waveguides. For the directional coupler in **Figure 1(b)**, the field transfer function can be expressed as [16]

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$$ \begin{bmatrix} E_{out-1} \\ E_{out-2} \end{bmatrix} = \begin{bmatrix} t & j\kappa \\ j\kappa & t \end{bmatrix} \begin{bmatrix} E_{in-1} \\ E_{in-2} \end{bmatrix}, \qquad (1) $$

where $j = \sqrt{-1}$, $t$ and $\kappa$ are the self-coupling and cross coupling coefficients, which satisfies the relation $t^2 + \kappa^2 = 1$ when assuming lossless coupling, $E_{in-1}$, $E_{in-2}$, $E_{out-1}$, and $E_{out-2}$ are the input and output optical fields right before and after the coupling region. For the connecting waveguide in **Figure 1(c)**, the field transfer function can be given by [23]

$$ T = ae^{j\varphi}, \qquad (2) $$

where $a = e^{-\alpha L/2}$ is the round-trip transmission factor, with $\alpha$ and $L$ denoting the power propagation loss factor and the waveguide length, respectively. In **Eq. (2)**, $\varphi = 2\pi n_g L / \lambda$ is the round-trip phase shift, with $n_g$ and $\lambda$ denoting the group index and the wavelength, respectively. For each directional coupler, two equations can be derived, and one equation can be obtained for each connecting waveguide. This results in $2 \times 4 + 1 \times 6 = 14$ equations in total for the device shown in **Figure 1(a)**, which includes four directional couplers and six connecting waveguides.

<table><caption>**Table 1. Definitions of structural parameters of the device in Figure 1(a) and the corresponding scattering matrix equations.**</caption><thead><tr><th></th><th>Structural Parameters<br>(i = 1 - 4)</th><th>Field transmission<br>coefficient<br>t<sub>i</sub></th><th>Field cross coupling<br>coefficient<br>κ<sub>i</sub></th></tr></thead><tbody><tr><td rowspan="2">Directional couplers</td><td rowspan="2">Scattering matrix equations</td><td>$$ \begin{bmatrix} E_3 \\ E_4 \end{bmatrix} = \begin{bmatrix} t_1 & j\kappa_1 \\ j\kappa_1 & t_1 \end{bmatrix} \begin{bmatrix} E_1 \\ E_2 \end{bmatrix}, \begin{bmatrix} E_7 \\ E_8 \end{bmatrix} = \begin{bmatrix} t_2 & j\kappa_2 \\ j\kappa_2 & t_2 \end{bmatrix} \begin{bmatrix} E_5 \\ E_6 \end{bmatrix}, $$</td><td></td></tr><tr><td>$$ \begin{bmatrix} E_{12} \\ E_{11} \end{bmatrix} = \begin{bmatrix} t_3 & j\kappa_3 \\ j\kappa_3 & t_3 \end{bmatrix} \begin{bmatrix} E_9 \\ E_{10} \end{bmatrix}, \begin{bmatrix} E_{15} \\ E_{16} \end{bmatrix} = \begin{bmatrix} t_4 & j\kappa_4 \\ j\kappa_4 & t_4 \end{bmatrix} \begin{bmatrix} E_{13} \\ E_{14} \end{bmatrix}. $$</td><td></td></tr><tr><td rowspan="2">Connecting waveguides</td><td rowspan="2">Structural Parameters<br>(i = 1 - 6)</td><td>Length</td><td>Transmission factor</td><td>Phase shift</td></tr><tr><td>L<sub>i</sub></td><td>a<sub>i</sub></td><td>φ<sub>i</sub></td></tr><tr><td colspan="2">Scattering matrix equations</td><td colspan="3">E<sub>5</sub> = T<sub>1</sub>E<sub>4</sub>, E<sub>9</sub> = T<sub>2</sub>E<sub>7</sub>, E<sub>13</sub> = T<sub>3</sub>E<sub>12</sub>,<br>E<sub>2</sub> = T<sub>4</sub>E<sub>15</sub>, E<sub>14</sub> = T<sub>5</sub>E<sub>8</sub>, E<sub>6</sub> = T<sub>6</sub>E<sub>16</sub>.</td></tr><tr><td colspan="2">Input</td><td colspan="3">E<sub>1</sub> = 1, E<sub>10</sub> = 0.</td></tr></tbody></table>

Third, the system input is set. For instance, if we assume that there is only a continuous-wave (CW) input from Port 1, then another two equations can be obtained: $E_1 = 1$ and $E_{10} = 0$. Here we set $E_1$ as 1 because the spectral transfer function at the output port, such as Port 2, is given by $f_{Port\ 2} = E_3 / E_1$. By setting $E_1$ to 1, $f_{Port\ 2} = E_3 / E_1 = E_3$, then the transfer function can be directly determined by calculating $E_3$ in the next step.

Finally, by solving all the linear equations obtained in the second and third steps, one can obtain the spectral transfer functions at the output ports. In **Table 1**, we summarize all the 16 equations for the device in **Figure 1(a)**. In these equations, there are 16 variables $E_i$ (i = 1–16), with the device’s structural parameters denoted using symbols and treated as constant coefficients. By solving this system of linear equations using computational tools (e.g., symbolic calculation in MATLAB), any of the variables $E_i$ (i = 1–16) can be determined as a function of the device’s structural parameters. For example, the spectral transfer function at Port 2 can be given by $f_{Port\ 2} = E_3 (L_i, t_i, \kappa_i)$. Assuming that the power propagation loss factor $\alpha$ and the group index $n_g$ are constant for the device, the spectral transfer function at Port 2 can be expressed as a function of $L_i, t_i$, and $\kappa_i$, i.e., $f_{Port\ 2} = E_3 (L_i, t_i, \kappa_i)$.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a) Calculated (i) intensity and (ii) phase response spectra at Port 2 for the device in **Figure 1** with different $t_1 = t_3$. (b) Calculated (i) intensity and (ii) phase response spectra at Port 3 for the device in **Figure 1** with different $t_1 = t_3$. In (a) and (b), $t_2 = t_4 = 0.98$. The power propagation loss factor $\alpha$ and group index $n_g$ are assumed to be 101 m⁻¹ and 4.335, respectively.</figcaption></figure>

In **Figure 2(a)**, we show the intensity and phase response spectra at Port 2 for different $t_1$ based on the obtained spectral transfer function $f_{\text{Port 2}} = E_2(L_i, t_i, \kappa_i)$. Except for the varying parameters, other structural parameters were chosen as follows: $L_1 = L_2 = L_3 = L_4 = 41.73$ µm, $L_5 = L_6 = 62.83$ µm, and $t_2 = t_4 = 0.98$. By using the same method, we obtain the intensity and phase responses at Port 3 for different $t_i$ as shown in **Figure 2(b)**. In our modeling, we assume that $\alpha = 101$ m⁻¹ and group index $n_g = 4.335$ based on values obtained from our previously fabricated silicon photonic devices [20, 25]. Unless otherwise specified, we use the same values for these two parameters in the modeling for all the devices in the following figures.

Our results in **Figure 2** show good agreement with the results in Ref. [26], confirming the effectiveness of our method. Since the obtained spectral transfer functions ($e.g., f_{\text{Port 2}} = E_2(L_i, t_i, \kappa_i$)) include all the structural parameters, the intensity and phase responses can be easily tailored by adjusting any of these parameters. This provides significant flexibility in designing and optimizing the spectral response of integrated photonic devices with different structural parameters. It should also be noted that our modeling, as mentioned above, is not limited to silicon photonic platforms. By using the corresponding values of $\alpha$ and $n_g$, it is applicable across various integrated platforms such as silicon nitride, doped silica, and lithium niobate [27-93].

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<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>(b-i)
Wavelength (nm)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(b-ii)
Wavelength (nm)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(c-i)
Wavelength (nm)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c-ii)
Wavelength (nm)</figcaption></figure>

**Figure 3.** (a) Schematic illustration of an integrated photonic resonator formed by an add-drop micro-ring resonator (AD-MRR) sandwiched between two Sagnac interferometers (SIs) with an input from Port 1. The device is divided into several directional couplers and connecting waveguides, with $E_i$ ($i$ = 1–16) denoting the optical fields at the dividing points. The electric fields propagating from left to right are defined as "+" direction and the ones propagating from right to left are defined as "-" direction. (b) Calculated (i) power transmission and (ii) phase response spectra at Port 2 for different $t_1 = t_4$. (c) Calculated (i) power reflection and (ii) phase response spectra at Port 1 for different $t_1 = t_4$.

**3. Modeling of devices with bidirectional light propagation**

For the device in **Figure 1(a)**, light propagates in only one direction along each waveguide. In more complex integrated photonic resonators with light propagating bidirectionally in the waveguides, the modeling becomes more complicated. In this section, we use the device configuration shown in **Figure 3(a)** as an example to show how to model a complex integrated photonic resonator with bidirectional light propagation based on the SMM. The device consists of an add-drop (AD) MRR sandwiched between a pair of

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Sagnac interferometers (SIs), where the SIs introduce back reflection and hence bidirectional light propagation in such device. 144

To model the device in Figure 3(a), we first divide it into several directional couplers and connecting waveguides – the same as what we did for the device in Figure 1(a). The optical fields at the dividing points between directional couplers and connecting waveguides are denoted as $E_i$ ($i$ = 1–16). Since there are optical fields traveling in two different directions at each dividing point, the optical fields traveling from left to right or in a clockwise direction are defined as “+”, and the opposite direction is defined as “-”. 145

In the second step, we establish 28 scattering matrix equations describing the relation between $E_i^+$ and $E_i^-$ at the input / output ports of the directional couplers and the connecting waveguides, as shown in Table 2. For each directional coupler, four equations can be derived, and two equations can be obtained for each connecting waveguide. Note that the number of equations is doubled due to the bidirectional light propagation in such device. 152

In the third step, we obtain another four equations by setting the system input. If we assume that there is only a CW input from Port 1, then the four equations are: $E_{1+}^1 = 1$, $E_{1-}^1 = 0$, $E_{10+}^1 = 0$, and $E_{16-}^1 = 0$. Here the number of equations is also doubled due to the bidirectional light propagation. 153

Finally, by solving all the 32 linear equations obtained previously (which include 32 variables, i.e., $E_i^+$ and $E_i^-$, $i$ = 1–16), we can obtain the spectral transfer functions at all the output ports. For example, the transfer function for forward transmission, with output at Port 2, is $f_{\text{Port 2}} = E_{16+}^1(L_i, t_i, \kappa_i)$, whereas the transfer function for backward reflection is $f_{\text{Port 1}} = E_{1-}^1(L_i, t_i, \kappa_i)$, with output at Port 1. It should be noted that as the number of equations increases or the device configuration becomes more complex, the use of computational software to solve the system of linear equations offers significant advantages as compared to manual derivation of the spectral transfer functions as those in Refs. [25, 26]. 160

<table><caption>Table 2. Definitions of structural parameters of the device in Figure 3(a) and the corresponding scattering matrix equations. 170</caption><thead><tr><th></th><th>Structural Parameters<br>(i = 1 - 4)</th><th colspan="2">Field transmission coefficient</th><th colspan="2">Field cross coupling coefficient</th></tr><tr><th></th><th></th><th colspan="2">t<sub>i</sub></th><th colspan="2">κ<sub>i</sub></th></tr></thead><tbody><tr><td rowspan="5">Directional couplers</td><td rowspan="5">Scattering matrix equations</td><td>$[E_2^+], [E_3^-] = [t_1, jκ_1, t_1]$</td><td>$[E_1^+], [E_4^-] = [t_1, jκ_1, t_1]$</td><td>$[E_1^+], [E_4^-] = [t_1, jκ_1, t_1]$</td><td>$[E_2^+], [E_3^-] = [t_1, jκ_1, t_1]$</td></tr><tr><td>$[E_7^+], [E_8^-] = [t_2, jκ_2, t_2]$</td><td>$[E_5^+], [E_6^-] = [t_2, jκ_2, t_2]$</td><td>$[E_5^+], [E_6^-] = [t_2, jκ_2, t_2]$</td><td>$[E_7^+], [E_8^-] = [t_2, jκ_2, t_2]$</td></tr><tr><td>$[E_{11}^+], [E_{12}^+] = [t_3, jκ_3, t_3]$</td><td>$[E_9^+], [E_{10}^+] = [t_3, jκ_3, t_3]$</td><td>$[E_9^+], [E_{10}^-] = [t_3, jκ_3, t_3]$</td><td>$[E_{11}^+], [E_{12}^-] = [t_3, jκ_3, t_3]$</td></tr><tr><td>$[E_{14}^+], [E_{15}^-] = [t_4, jκ_4, t_4]$</td><td>$[E_{13}^+], [E_{16}^-] = [t_4, jκ_4, t_4]$</td><td>$[E_{13}^+], [E_{16}^-] = [t_4, jκ_4, t_4]$</td><td>$[E_{14}^+], [E_{15}^-] = [t_4, jκ_4, t_4]$</td></tr><tr><td>$[E_3^+], [E_2^-] = T_1 [E_2^+, E_3^-]$</td><td>$[E_5^+], [E_4^-] = T_2 [E_4^+, E_5^-]$</td><td>$[E_5^+], [E_8^-] = T_3 [E_8^+, E_9^-]$</td><td>$[E_6^+], [E_{11}^+, E_{12}^-] = T_4 [E_{11}^+, E_{12}^-]$</td></tr><tr><td rowspan="2">Connecting waveguides</td><td rowspan="2">Scattering matrix equations</td><td>$[E_6^+], [E_{11}^-] = T_4 [E_{11}^+, E_6^-]$</td><td>$[E_{13}^+], [E_{12}^-] = T_5 [E_{12}^+, E_{13}^-]$</td><td>$[E_{13}^+], [E_{14}^-] = T_6 [E_{14}^+, E_{15}^-]$</td><td></td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td colspan="2">Input</td><td colspan="4">$E_{1+}^1 = 1$, $E_{1-}^1 = 0$, $E_{10+}^1 = 0$, $E_{16-}^1 = 0$. 172</td></tr></tbody></table>

Based on the obtained spectral transfer functions at Port 2 and Port 1, we plot the corresponding response spectra for different $t_i$, as shown in Figures 3(b) and 3(c), respectively. Except for $t_i$, the other structural parameters are kept as constant as: $L_1 = L_6 = 129.66$ µm, $L_2 = L_5 = 77.67$ µm, $L_3 = L_4 = 63.33$ µm, and $t_2 = t_3 = 0.95$. Our results in Figure 3 also
show good agreement with the results in Ref. [25], confirming the effectiveness of our method in modeling complex integrated photonic resonator with bidirectional light propagation. 177 178 179

### 4. Simplification by dividing device into submodules 180

In modeling of the device configurations in Figures 1(a) and 3(a), the devices were divided into several basic elements including directional couplers and connecting waveguides. To reduce the number of equations and simplify the calculation of the spectral transfer function, the device configuration can also be divided into several independent submodules such as MRRs and SIs, which are formed by the basic elements. 181 182 183 184 185 186

<figure><img src="image_11.png" /><figcaption>**Figure 4.** Dividing device configuration into submodules to simplify the calculation (a) Device configuration in Figure 1(a), where the add-drop micro-ring resonator (AD-MRR) at the center is regarded as an independent module. (b) Device configuration in Figure 3(a), which is divided into submodules including an AD-MRR, two Sagnac interferometers (SIs), and connecting waveguides between them. 187 188 189 190 191 192 193</figcaption></figure>

As illustrated in Figure 4(a), the AD-MRR at the center of the device in Figure 1(a) is treated as an independent module, with the field transfer functions at the through and drop ports denoted as $T_{RR}$ and $D_{RR}$, respectively. This results in 12 equations in total obtained by using our method mentioned previously, as summarized in Table 3. In comparison to the 16 equations obtained for the same device configuration in Table 1, the number of equations is reduced, which helps to simplify the solving of the system of equations. By solving the 12 equations in Table 3, the spectral transfer function at Port 2 can be obtained, which is a function of $T_{MRR}$ and $D_{MRR}$. We compared the response spectra obtained by solving the equations in Tables 1 and 3 and found them to be identical. This verifies the validity of the new dividing method in Figures 4(a). It is also worth mentioning that when plotting the response spectra based on this dividing method, $T_{MRR}$ and $D_{MRR}$ should be further expressed as [94] 194 195 196 197 198 199 200 201 202 203 204 205
$$T_{RR} = \frac{E_6}{E_5} = \frac{E_{12}}{E_{11}} = \frac{t_{RR-1} - t_{RR-2}a_{RR}e^{j\phi_{RR}}}{1 - t_{RR-1}t_{RR-2}a_{RR}e^{j\phi_{RR}}}, \quad (3)$$

$$D_{RR} = \frac{E_{12}}{E_5} = \frac{E_6}{E_{11}} = \frac{-\kappa_{RR-1}\kappa_{RR-2}\sqrt{a_{RR}}e^{j\phi_{RR}}}{1 - t_{RR-1}t_{RR-2}a_{RR}e^{j\phi_{RR}}}, \quad (4)$$

where $j = \sqrt{-1}$, $a_{RR}$ and $\phi_{RR}$ are the round-trip transmission factor and phase shift along the ring in the AD-MRR, respectively. $t_{RR-i}$ and $\kappa_{RR-i}$ ($i = 1, 2$) are the self-coupling and cross coupling coefficients of the directional couplers in the AD-MRR, respectively.

<table><caption>**Table 3.** Scattering matrix equations for the device in Figure 4(a). $T_{RR}$ and $D_{RR}$ are the field transfer functions at the through and drop ports for the AD-MRR, respectively.</caption><tbody><tr><td rowspan="3">Scattering matrix equations</td><td>Add-drop micro-ring resonator (AD-MRR)</td><td>$\begin{bmatrix} E_6 \\ E_{12} \end{bmatrix} = \begin{bmatrix} T_{RR} & D_{RR} \\ D_{RR} & T_{RR} \end{bmatrix} \begin{bmatrix} E_5 \\ E_{11} \end{bmatrix}$</td></tr><tr><td>Directional couplers</td><td>$\begin{bmatrix} E_3 \\ E_4 \end{bmatrix} = \begin{bmatrix} t_1 & j\kappa_1 \\ j\kappa_1 & t_1 \end{bmatrix} \begin{bmatrix} E_1 \\ E_2 \end{bmatrix}, \begin{bmatrix} E_{10} \\ E_9 \end{bmatrix} = \begin{bmatrix} t_2 & j\kappa_2 \\ j\kappa_2 & t_2 \end{bmatrix} \begin{bmatrix} E_7 \\ E_8 \end{bmatrix}$</td></tr><tr><td>Connecting waveguides</td><td>$E_5 = T_{w1}E_4, E_7 = T_{w2}E_6, E_{11} = T_{w3}E_{10}, E_2 = T_{w4}E_{12}$</td></tr><tr><td>Input</td><td colspan="2">$E_1 = 1, E_8 = 0$.</td></tr></tbody></table>

Similarly, in **Figure 4(b)** we divided the device configuration in **Figure 3(a)** into an add-drop MRR, two SIs, and several connecting waveguides between them. By using this new dividing method, we obtained the same response spectra as those in **Figures 3(b) and 3(c)**. When plotting the response spectra, the forward and backward field transfer functions for the SIs were further expressed as

$$T_{SI-i} = \frac{E_2^+}{E_1^+} = \frac{E_6^+}{E_5^+} = (t_{SI-i}^2 - \kappa_{SI-i}^2)a_{SI-i}e^{-j\phi_{SI-i}}, \quad (i=1, 2) \quad (5)$$

$$R_{SI-i} = \frac{E_1}{E_1^+} = \frac{E_5}{E_5^+} = 2jt_{SI-i}\kappa_{SI-i}a_{SI-i}e^{-j\phi_{SI-i}}, \quad (i=1, 2) \quad (6)$$

where $j = \sqrt{-1}$, $a_{SI-i}$ and $\phi_{SI-i}$ ($i = 1, 2$) are the round-trip transmission factor and phase shift along rings in the SIs, respectively. $t_{SI-i}$ and $\kappa_{SI-i}$ ($i = 1, 2$) are the self-coupling and cross coupling coefficients of the directional couplers in the SIs, respectively.

As shown in **Table 4**, only 10 equations are established by using the new dividing method, in contrast to 32 equations obtained for the dividing method in **Figure 3(a)**. This indicates that although dividing device into basic elements is a universal approach, dividing it into submodules simplifies the process of solving equations. This is particularly true for devices composed of multiple submodules, where the advantages of the new division method become even more evident.

<table><caption>**Table 4.** Scattering matrix equations for the device in Figure 4(b). $T_{SI-i}$ and $R_{SI-i}$ ($i = 1, 2$) are forward and backward field transfer functions for the SIs, respectively. $T_{RR}$ and $D_{RR}$ are the field transfer functions at the through and drop ports for the AD-MRR, respectively.</caption><tbody><tr><td rowspan="3">Scattering matrix equations</td><td>Sagnac interferometers (SIs)</td><td>$\begin{bmatrix} E_2^+ \\ E_1^- \end{bmatrix} = \begin{bmatrix} T_{SI-1} & R_{SI-1} \\ R_{SI-1} & T_{SI-1} \end{bmatrix} \begin{bmatrix} E_1^+ \\ E_2^- \end{bmatrix}, \begin{bmatrix} E_6^+ \\ E_5^- \end{bmatrix} = \begin{bmatrix} T_{SI-2} & R_{SI-2} \\ R_{SI-2} & T_{SI-2} \end{bmatrix} \begin{bmatrix} E_5^+ \\ E_6^- \end{bmatrix}$</td></tr><tr><td>Add-drop micro-ring resonator (AD-MRR)</td><td>$\begin{bmatrix} E_4^+ \\ E_3^- \end{bmatrix} = D_{RR} \begin{bmatrix} E_3^+ \\ E_4^- \end{bmatrix}$</td></tr><tr><td>Connecting waveguides</td><td>$\begin{bmatrix} E_3^+ \\ E_2^- \end{bmatrix} = T_{w1} \begin{bmatrix} E_2^+ \\ E_3^- \end{bmatrix}, \begin{bmatrix} E_5^+ \\ E_4^- \end{bmatrix} = T_{w2} \begin{bmatrix} E_4^+ \\ E_5^- \end{bmatrix}$</td></tr><tr><td>Input</td><td colspan="2">$E_1^+ = 1, E_6^- = 0$.</td></tr></tbody></table>
<figure><img src="image_12.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b-i) Wavelength (nm)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(c-i) Wavelength (nm)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(c-ii) Wavelength (nm)</figcaption></figure>

**Figure 5.** (a) Schematic illustration of an integrated photonic resonator formed by eight cascaded Sagnac interferometers (SIs) with input from Port 1. The device is divided into eight SIs and connecting waveguides between them, with $E_i$ ($i = 1–16$) denoting the optical fields at the dividing points. The electric fields propagating from to right to left are defined as "+" direction and the ones propagating from left to right are defined as "-" direction. (b) Calculated (i) power transmission and (ii) phase response spectra at Port 2 for different $t_i$ ($i = 1–8$), which are the self-coupling coefficients of the directional couplers in SIs. (c) Calculated (i) power reflection and (ii) phase response spectra at Port 1 for different $t_i$ ($i = 1–8$). In (b) and (c), the circumference of the rings in the SIs are 129.66 µm, and the lengths of the connecting waveguides between them are 100 µm.

In **Figure 5**, we model a more complex device by dividing it into eight SIs and establishing 32 linear equations, as summarized in **Table 5**. The forward and backward field transfer functions for the SIs are denoted as $T_{SI-i}$ and $R_{SI-i}$ ($i = 1–8$), respectively. Dividing the device into basic elements would result in a system of 64 equations, compared to the 32 equations shown in **Table 5**. By solving the system of 32 equations to obtain $E_{16+}$ and $E_{1-}$, we obtain the spectral transfer functions at Port 2 and Port 1 (reflection), respectively. The corresponding response spectra are plotted in **Figures 5(b) and 5(c)**, which are consistent with the results in Ref. [95] and further confirms the effectiveness of our method in modeling devices with complex configurations.

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<table><caption>Table 5. Scattering matrix equations for the device in Figure 5(a). $T_{SI-i}$ and $R_{SI-i}$ ($i = 1-8$) are the forward and backward field transfer functions for the SIs, respectively.</caption><tbody><tr><td rowspan="4">Scattering matrix equations</td><td rowspan="4">Sagnac interferometers (SIs)</td><td>$\begin{bmatrix} E_2^+ \\ E_1^- \end{bmatrix} = \begin{bmatrix} T_{SI-1} & R_{SI-1} \\ R_{SI-1} & T_{SI-1} \end{bmatrix} \begin{bmatrix} E_1^+ \\ E_2^- \end{bmatrix}, \begin{bmatrix} E_4^+ \\ E_3^- \end{bmatrix} = \begin{bmatrix} T_{SI-2} & R_{SI-2} \\ R_{SI-2} & T_{SI-2} \end{bmatrix},$</td></tr><tr><td>$\begin{bmatrix} E_6^+ \\ E_5^- \end{bmatrix} = \begin{bmatrix} T_{SI-3} & R_{SI-3} \\ R_{SI-3} & T_{SI-3} \end{bmatrix} \begin{bmatrix} E_5^+ \\ E_6^- \end{bmatrix}, \begin{bmatrix} E_8^+ \\ E_7^- \end{bmatrix} = \begin{bmatrix} T_{SI-4} & R_{SI-4} \\ R_{SI-4} & T_{SI-4} \end{bmatrix} \begin{bmatrix} E_7^+ \\ E_8^- \end{bmatrix},$</td></tr><tr><td>$\begin{bmatrix} E_{10}^+ \\ E_9^- \end{bmatrix} = \begin{bmatrix} T_{SI-5} & R_{SI-5} \\ R_{SI-5} & T_{SI-5} \end{bmatrix} \begin{bmatrix} E_9^+ \\ E_{10}^- \end{bmatrix}, \begin{bmatrix} E_{12}^+ \\ E_{11}^- \end{bmatrix} = \begin{bmatrix} T_{SI-6} & R_{SI-6} \\ R_{SI-6} & T_{SI-6} \end{bmatrix} \begin{bmatrix} E_{11}^+ \\ E_{12}^- \end{bmatrix},$</td></tr><tr><td>$\begin{bmatrix} E_{14}^+ \\ E_{13}^- \end{bmatrix} = \begin{bmatrix} T_{SI-7} & R_{SI-7} \\ R_{SI-7} & T_{SI-7} \end{bmatrix} \begin{bmatrix} E_{13}^+ \\ E_{14}^- \end{bmatrix}, \begin{bmatrix} E_{16}^+ \\ E_{15}^- \end{bmatrix} = \begin{bmatrix} T_{SI-8} & R_{SI-8} \\ R_{SI-8} & T_{SI-8} \end{bmatrix} \begin{bmatrix} E_{15}^+ \\ E_{16}^- \end{bmatrix}.$</td></tr><tr><td rowspan="4">Connecting waveguides</td><td rowspan="4"></td><td>$\begin{bmatrix} E_3^+ \\ E_2^- \end{bmatrix} = T_{w1} \begin{bmatrix} E_2^+ \\ E_3^- \end{bmatrix}, \begin{bmatrix} E_5^+ \\ E_4^- \end{bmatrix} = T_{w2} \begin{bmatrix} E_4^+ \\ E_5^- \end{bmatrix}, \begin{bmatrix} E_7^+ \\ E_6^- \end{bmatrix} = T_{w3} \begin{bmatrix} E_6^+ \\ E_7^- \end{bmatrix},$</td></tr><tr><td>$\begin{bmatrix} E_9^+ \\ E_8^- \end{bmatrix} = T_{w4} \begin{bmatrix} E_8^+ \\ E_9^- \end{bmatrix}, \begin{bmatrix} E_{11}^+ \\ E_{10}^- \end{bmatrix} = T_{w5} \begin{bmatrix} E_{10}^+ \\ E_{11}^- \end{bmatrix}, \begin{bmatrix} E_{13}^+ \\ E_{12}^- \end{bmatrix} = T_{w6} \begin{bmatrix} E_{12}^+ \\ E_{13}^- \end{bmatrix},$</td></tr><tr><td>$\begin{bmatrix} E_{15}^+ \\ E_{14}^- \end{bmatrix} = T_{w7} \begin{bmatrix} E_{14}^+ \\ E_{15}^- \end{bmatrix}.$</td></tr><tr><td colspan="2">Input</td></tr><tr><td colspan="2">$E_{16}^+ = 1, E_{1}^- = 0.$</td></tr></tbody></table>

When dividing complex devices into submodules, it's important to ensure that the submodules are independent. Here, "independent" means that the submodules exchange energy with other parts exclusively through connecting waveguides. For the device shown in **Figure 6a** formed by a self-coupled waveguide, even though there are multiple SIs within the device, it cannot be divided into submodules of SIs. This is because the adjacent SIs are mutually coupled with energy exchange through directional couplers. In this case, the spectral transfer functions can be obtained by using the universal dividing method, which divides the device into basic elements of directional couplers and connecting waveguides.

<figure><img src="image_16.png" /><figcaption>**Figure 6.** (a) Schematic illustration of an integrated photonic resonator formed by a self-coupled waveguide with input from Port 1. The device is divided into several directional couplers and connecting waveguides, with $E_i$ ($i = 1-28$) denoting the optical fields at the dividing points. The electric fields propagating from to up to down are defined as "+" direction and the ones propagating from down to up are defined as "-" direction. (b) Calculated (i) power transmission and (ii) phase response spectra at Port 2 for different $t_i$, which are the are the self-coupling coefficients of the directional couplers. (c) Calculated (i) power reflection and (ii) phase response spectra at Port 1 for different $t_i$.</figcaption></figure>

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**Table 6** shows a system of 56 equations obtained by using the universal dividing method for the device in **Figure 6a**. By solving these equations to obtain $E_{28^+}$ and $E_{1^-}$, we obtain the spectral transfer functions at Port 1 and Port 2, respectively. The corresponding response spectra for different $t_i$ are shown in **Figures 6(b) and 6(c)**. Except for $t_i$, all other structural parameters are kept constant as: $L_1 = 180 \ \mu\text{m}$, $L_2 = 60 \ \mu\text{m}$ and $L_3 = 120 \ \mu\text{m}$. The response spectra also show a good agreement with the results in Ref. [24], providing additional evidence of the effectiveness of our method.

<table><caption>**Table 6. Definitions of structural parameters of the device in Figure 6(a) and the corresponding scattering matrix equations.**</caption><thead><tr><th></th><th>Structural<br>Parameters<br>(i = 1 – 7)</th><th>Field transmission coefficient<br>t<sub>i</sub></th><th>Field cross coupling coefficient<br>κ<sub>i</sub></th></tr></thead><tbody><tr><td rowspan="10">Directional<br>couplers</td><td rowspan="10">Scattering<br>matrix<br>equations</td><td>$\begin{bmatrix} E_2^+ \\ E_5^- \end{bmatrix} = \begin{bmatrix} t_1 & j\kappa_1 \\ j\kappa_1 & t_1 \end{bmatrix} \begin{bmatrix} E_1^+ \\ E_6^- \end{bmatrix}, \begin{bmatrix} E_1^+ \\ E_6^- \end{bmatrix} = \begin{bmatrix} t_1 & j\kappa_1 \\ j\kappa_1 & t_1 \end{bmatrix} \begin{bmatrix} E_2^- \\ E_5^+ \end{bmatrix}$</td><td></td></tr><tr><td>$\begin{bmatrix} E_4^+ \\ E_9^- \end{bmatrix} = \begin{bmatrix} t_2 & j\kappa_2 \\ j\kappa_2 & t_2 \end{bmatrix} \begin{bmatrix} E_3^+ \\ E_{10}^- \end{bmatrix}, \begin{bmatrix} E_3^+ \\ E_{10}^- \end{bmatrix} = \begin{bmatrix} t_2 & j\kappa_2 \\ j\kappa_2 & t_2 \end{bmatrix} \begin{bmatrix} E_4^- \\ E_9^+ \end{bmatrix}$</td><td></td></tr><tr><td>$\begin{bmatrix} E_8^+ \\ E_{13}^- \end{bmatrix} = \begin{bmatrix} t_3 & j\kappa_3 \\ j\kappa_3 & t_3 \end{bmatrix} \begin{bmatrix} E_7^+ \\ E_{14}^- \end{bmatrix}, \begin{bmatrix} E_7^+ \\ E_{14}^- \end{bmatrix} = \begin{bmatrix} t_3 & j\kappa_3 \\ j\kappa_3 & t_3 \end{bmatrix} \begin{bmatrix} E_8^- \\ E_{13}^+ \end{bmatrix}$</td><td></td></tr><tr><td>$\begin{bmatrix} E_{12}^+ \\ E_{17}^- \end{bmatrix} = \begin{bmatrix} t_4 & j\kappa_4 \\ j\kappa_4 & t_4 \end{bmatrix} \begin{bmatrix} E_{11}^+ \\ E_{18}^- \end{bmatrix}, \begin{bmatrix} E_{18}^+ \\ E_{11}^- \end{bmatrix} = \begin{bmatrix} t_4 & j\kappa_4 \\ j\kappa_4 & t_4 \end{bmatrix} \begin{bmatrix} E_{12}^- \\ E_{17}^+ \end{bmatrix}$</td><td></td></tr><tr><td>$\begin{bmatrix} E_{16}^+ \\ E_{21}^- \end{bmatrix} = \begin{bmatrix} t_5 & j\kappa_5 \\ j\kappa_5 & t_5 \end{bmatrix} \begin{bmatrix} E_{15}^+ \\ E_{22}^- \end{bmatrix}, \begin{bmatrix} E_{15}^+ \\ E_{22}^- \end{bmatrix} = \begin{bmatrix} t_5 & j\kappa_5 \\ j\kappa_5 & t_5 \end{bmatrix} \begin{bmatrix} E_{16}^- \\ E_{21}^+ \end{bmatrix}$</td><td></td></tr><tr><td>$\begin{bmatrix} E_{20}^+ \\ E_{25}^- \end{bmatrix} = \begin{bmatrix} t_6 & j\kappa_6 \\ j\kappa_6 & t_6 \end{bmatrix} \begin{bmatrix} E_{19}^+ \\ E_{26}^- \end{bmatrix}, \begin{bmatrix} E_{19}^+ \\ E_{26}^- \end{bmatrix} = \begin{bmatrix} t_6 & j\kappa_6 \\ j\kappa_6 & t_6 \end{bmatrix} \begin{bmatrix} E_{20}^- \\ E_{25}^+ \end{bmatrix}$</td><td></td></tr><tr><td>$\begin{bmatrix} E_{24}^+ \\ E_{27}^- \end{bmatrix} = \begin{bmatrix} t_7 & j\kappa_7 \\ j\kappa_7 & t_7 \end{bmatrix} \begin{bmatrix} E_{23}^+ \\ E_{28}^- \end{bmatrix}, \begin{bmatrix} E_{23}^+ \\ E_{28}^- \end{bmatrix} = \begin{bmatrix} t_7 & j\kappa_7 \\ j\kappa_7 & t_7 \end{bmatrix} \begin{bmatrix} E_{24}^- \\ E_{27}^+ \end{bmatrix}$</td><td></td></tr><tr><td></td><td>Structural<br>Parameters<br>(i = 1 – 3)</td><td>Length<br>L<sub>i</sub></td><td>Transmission factor<br>a<sub>i</sub></td><td>Phase shift<br>φ<sub>i</sub></td></tr><tr><td rowspan="5">Connecting<br>waveguides</td><td rowspan="5">Scattering<br>matrix<br>equations</td><td>$\begin{bmatrix} E_3^+ \\ E_2^- \end{bmatrix} = T_1 \begin{bmatrix} E_2^+ \\ E_3^- \end{bmatrix}, \begin{bmatrix} E_5^+ \\ E_4^- \end{bmatrix} = T_2 \begin{bmatrix} E_4^+ \\ E_5^- \end{bmatrix}, \begin{bmatrix} E_7^+ \\ E_6^- \end{bmatrix} = T_3 \begin{bmatrix} E_6^+ \\ E_7^- \end{bmatrix}, \begin{bmatrix} E_9^+ \\ E_8^- \end{bmatrix} = T_2 \begin{bmatrix} E_8^+ \\ E_9^- \end{bmatrix}$</td><td></td><td></td></tr><tr><td>$\begin{bmatrix} E_{11}^+ \\ E_{10}^- \end{bmatrix} = T_3 \begin{bmatrix} E_{10}^+ \\ E_{11}^- \end{bmatrix}, \begin{bmatrix} E_{13}^+ \\ E_{12}^- \end{bmatrix} = T_2 \begin{bmatrix} E_{12}^+ \\ E_{13}^- \end{bmatrix}, \begin{bmatrix} E_{15}^+ \\ E_{14}^- \end{bmatrix} = T_3 \begin{bmatrix} E_{14}^+ \\ E_{15}^- \end{bmatrix}$</td><td></td><td></td></tr><tr><td>$\begin{bmatrix} E_{17}^+ \\ E_{16}^- \end{bmatrix} = T_2 \begin{bmatrix} E_{16}^+ \\ E_{17}^- \end{bmatrix}, \begin{bmatrix} E_{19}^+ \\ E_{18}^- \end{bmatrix} = T_3 \begin{bmatrix} E_{18}^+ \\ E_{19}^- \end{bmatrix}, \begin{bmatrix} E_{21}^+ \\ E_{20}^- \end{bmatrix} = T_2 \begin{bmatrix} E_{20}^+ \\ E_{21}^- \end{bmatrix}$</td><td></td><td></td></tr><tr><td>$\begin{bmatrix} E_{23}^+ \\ E_{22}^- \end{bmatrix} = T_3 \begin{bmatrix} E_{22}^+ \\ E_{23}^- \end{bmatrix}, \begin{bmatrix} E_{25}^+ \\ E_{24}^- \end{bmatrix} = T_2 \begin{bmatrix} E_{24}^+ \\ E_{25}^- \end{bmatrix}, \begin{bmatrix} E_{27}^+ \\ E_{26}^- \end{bmatrix} = T_1 \begin{bmatrix} E_{26}^+ \\ E_{27}^- \end{bmatrix}$</td><td></td><td></td></tr><tr><td></td><td>Input</td><td></td><td></td><td></td></tr></tbody><tfoot><tr><td colspan="5">E<sub>1</sub><sup>+</sup> = 1, E<sub>28</sub><sup>-</sup> = 0.</td></tr></tfoot></table>

**5. Deviations induced by approximations**

As evidenced by the results in previous sections, our method proves effective for modeling integrated photonic resonators and shows advantages for modeling those with complex configurations. Despite this, the method still has limitations that could cause deviations between the simulation results and measurements for practical devices. In this section, we discuss the limitations of our method to model integrated photonic resonators and methods for improving modeling accuracy.

In our modeling, we assume the self-coupling coefficient $t$ and cross-coupling coefficient $\kappa$ of the directional couplers to be wavelength-independent constant, while this holds true only within a certain wavelength range. According to Ref. [34 95 32], the field cross-coupling coefficient $\kappa$ of a directional coupler with a coupling length of $L_c$ can be given by
$$ \kappa = \sin\left(\frac{\pi}{2} \cdot \frac{L_c}{L_x}\right), \qquad (7) $$

where $L_x$ is the cross-over length, defined as the minimum distance at which optical power completely transfers from one waveguide to the other. The $L_x$ can be further expressed as [95]

$$ L_x = \frac{\lambda}{2(n_{\text{eff, even}} - n_{\text{eff, odd}})}. \qquad (8) $$

where $\lambda$ is the light wavelength, $n_{\text{eff, even}}$ and $n_{\text{eff, odd}}$ are the effective indices of the two fundamental eigenmodes of the coupled waveguides, respectively. From Eqs. (7) and (8), it can be seen that the coupling strength of the directional coupler is actually wavelength dependent due to the existence of dispersion (which leads to wavelength-dependent values of $n_{\text{eff, even}}$ and $n_{\text{eff, odd}}$). Therefore, the variation in the coupling strength of the directional couplers with wavelength can no longer be ignored when modeling the device's response spectra over a broad bandwidth (typically > 30 nm).

Another potential limitation arises from the fact that we use a constant waveguide group index $n_g$ instead of wavelength-dependent waveguide effective index $n_{\text{eff}}$ when calculating the phase shift along waveguides. In fact, the relation between $n_g$ and $n_{\text{eff}}$ can be expressed as [94]:

$$ n_g = n_{\text{eff}}(\lambda) - \lambda \frac{dn_{\text{eff}}(\lambda)}{d\lambda}, \qquad (9) $$

where $n_{\text{eff}}(\lambda)$ is the effective index as a function of light wavelength $\lambda$. The group index $n_g$ is widely used for calculating the free spectral ranges (FSRs) of integrated photonic resonators. For example, the FSR for a MRR can be approximately given by [94]

$$ \lambda_{\text{FSR}} \approx \frac{\lambda_0^2}{n_g L}, \qquad (10) $$

where $\lambda_0$ is the resonance wavelength and $L$ is the ring circumference.

Similar to the cross-coupling coefficient $\kappa$ of a directional coupler, the group index $n_g$ can be regarded as a constant coefficient only within a specific wavelength range. In contrast to introducing $n_{\text{eff}}$ at different wavelengths, using a constant $n_g$ can greatly simplify the plotting of the response spectra based on the spectral transfer function. However, this approximation only holds for modeling spectral response over a relatively narrow wavelength range (e.g., < 30 nm). When modeling the spectral response over a broad bandwidth or for nonlinear optical devices that are sensitive to dispersion-induced phase mismatch [8, 96-128], the wavelength-dependent $n_{\text{eff}}$ should be used to achieve more accurate simulation results.

## 6. Conclusion

In summary, a universal approach for modeling integrated photonic resonators with complex structures is proposed based on the SMM. By dividing the device configuration into basic elements such as directional couplers and connecting waveguides, our approach shows effectiveness in modeling integrated photonic resonators with both unidirectional and bidirectional light propagation, with the simulated spectral response agreeing well with experimental results. By dividing the device configuration into independent submodules MRRs and SIs, the modeling in our approach can also be simplified. Finally, we discuss the limitations arising from approximations in our modeling and the corresponding strategies for improving modeling accuracy. Our approach offers an efficient way for designing and optimizing complex integrated photonic devices, which is applicable across a wide range of integrated platforms. [129-147]

**Conflicts of Interest:** The authors declare no conflicts of interest.
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84. Yunping Bai, Xingyuan Xu,1, Mengxi Tan, Yang Sun, Yang Li, Jiayang Wu, Roberto Morandotti, Arnan Mitchell, Kun Xu, and David J. Moss, "Photonic multiplexing techniques for neuromorphic computing", Nanophotonics vol. 12 (5): 795-817 (2023). DOI:10.1515/nanoph-2022-0485. 520

85. Chawaphon Prayoonyong, Andreas Boes, Xingyuan Xu, Mengxi Tan, Sai T. Chu, Brent E. Little, Roberto Morandotti, Arnan Mitchell, David J. Moss, and Bill Corcoran, "Frequency comb distillation for optical superchannel transmission", Journal of Lightwave Technology vol. 39 (23) 7383-7392 (2021). DOI: 10.1109/JLT.2021.3116614. 523

86. Mengxi Tan, Xingyuan Xu, Jiayang Wu, Bill Corcoran, Andreas Boes, Thach G. Nguyen, Sai T. Chu, Brent E. Little, Roberto Morandotti, Arnan Mitchell, and David J. Moss, "Integral order photonic RF signal processors based on a soliton crystal micro-comb source", IOP Journal of Optics vol. 23 (11) 125701 (2021). https://doi.org/10.1088/2040-8986/ac2eab 526

87. Yang Sun, Jiayang Wu, Yang Li, Xingyuan Xu, Guanghui Ren, Mengxi Tan, Sai Tak Chu, Brent E. Little, Roberto Morandotti, Arnan Mitchell, and David J. Moss, "Optimizing the performance of microcomb based microwave photonic 530
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104. Yang Qu, Jiayang Wu, Yuning Zhang, Yunyi Yang, Linnan Jia, Baohua Jia, and David J. Moss, "Photo thermal tuning in GO-coated integrated waveguides", Micromachines Vol. 13 1194 (2022). doi.org/10.3390/mi13081194

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105. Yuning Zhang, Jiayang Wu, Yunyi Yang, Yang Qu, Houssein El Dirani, Romain Crochemore, Corrado Sciancalepore, Pierre Demongodin, Christian Grillet, Christelle Monat, Baohua Jia, and David J. Moss, "Enhanced self-phase modulation in silicon nitride waveguides integrated with 2D graphene oxide films", IEEE Journal of Selected Topics in Quantum Electronics Vol. 29 (1) 5100413 (2023). DOI: 10.1109/JSTQE.2022.3177385

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106. Yuning Zhang, Jiayang Wu, Yunyi Yang, Yang Qu, Linnan Jia, Baohua Jia, and David J. Moss, "Enhanced spectral broadening of femtosecond optical pulses in silicon nanowires integrated with 2D graphene oxide films", Micromachines Vol. 13 756 (2022). DOI:10.3390/mi13050756.

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107. Jiayang Wu, Yuning Zhang, Junkai Hu, Yunyi Yang, Di Jin, Wenbo Liu, Duan Huang, Baohua Jia, David J. Moss, "Novel functionality with 2D graphene oxide films integrated on silicon photonic chips", Advanced Materials Vol. 36 2403659 (2024). DOI: 10.1002/adma.202403659.

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108. Di Jin, Jiayang Wu, Junkai Hu, Wenbo Liu1, Yuning Zhang, Yunyi Yang, Linnan Jia, Duan Huang, Baohua Jia, and David J. Moss, "Silicon photonic waveguide and microring resonator polarizers incorporating 2D graphene oxide films", Applied Physics Letters vol. 125, 000000 (2024); doi: 10.1063/5.0221793.

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109. Yuning Zhang, Jiayang Wu, Linnan Jia, Di Jin, Baohua Jia, Xiaoyong Hu, David Moss, Qihuang Gong, "Advanced optical polarizers based on 2D materials", npj Nanophotonics Vol. 1, (2024). DOI: 10.1038/s44310-024-00028-3.

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110. Junkai Hu, Jiayang Wu, Wenbo Liu, Di Jin, Houssein El Dirani, Sébastien Kerdiles, Corrado Sciancalepore, Pierre Demongodin, Christian Grillet, Christelle Monat, Duan Huang, Baohua Jia, and David J. Moss, "2D graphene oxide: a versatile thermo-optic material", Advanced Functional Materials 34 2406799 (2024). DOI: 10.1002/adfm.202406799.

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111. Yang Qu, Jiayang Wu, Yuning Zhang, Yunyi Yang, Linnan Jia, Houssein El Dirani, Sébastien Kerdiles, Corrado Sciancalepore, Pierre Demongodin, Christian Grillet, Christelle Monat, Baohua Jia, and David J. Moss, "Integrated optical parametric amplifiers in silicon nitride waveguides incorporated with 2D graphene oxide films", Light: Advanced Manufacturing 4 39 (2023). https://doi.org/10.37188/lam.2023.039.

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112. Di Jin, Wenbo Liu, Linnan Jia, Junkai Hu, Duan Huang, Jiayang Wu, Baohua Jia, and David J. Moss, "Thickness and Wavelength Dependent Nonlinear Optical Absorption in 2D Layered MXene Films", Small Science 4 2400179 (2024). DOI:10.1002/smsc202400179;

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113. Linnan Jia, Jiayang Wu, Yuning Zhang, Yang Qu, Baohua Jia, Zhigang Chen, and David J. Moss, "Fabrication Technologies for the On-Chip Integration of 2D Materials", Small: Methods Vol. 6, 2101435 (2022). DOI:10.1002/smtd.202101435.

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114. Yuning Zhang, Jiayang Wu, Yang Qu, Linnan Jia, Baohua Jia, and David J. Moss, "Design and optimization of four-wave mixing in microring resonators integrated with 2D graphene oxide films", Journal of Lightwave Technology Vol. 39 (20) 6553-6562 (2021). DOI:10.1109/JLT.2021.3101292.

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115. Yuning Zhang, Jiayang Wu, Yang Qu, Linnan Jia, Baohua Jia, and David J. Moss, "Optimizing the Kerr nonlinear optical performance of silicon waveguides integrated with 2D graphene oxide films", Journal of Lightwave Technology Vol. 39 (14) 4671-4683 (2021). DOI: 10.1109/JLT.2021.3069733.

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116. Yang Qu, Jiayang Wu, Yuning Zhang, Yao Liang, Baohua Jia, and David J. Moss, "Analysis of four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films", Journal of Lightwave Technology Vol. 39 (9) 2902-2910 (2021). DOI: 10.1109/JLT.2021.3059721.

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117. Jiayang Wu, Linnan Jia, Yuning Zhang, Yang Qu, Baohua Jia, and David J. Moss, "Graphene oxide: versatile films for flat optics to nonlinear photonic chips", Advanced Materials Vol. 33 (3) 2006415, pp.1-29 (2021). DOI:10.1002/adma.202006415.

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118. Y. Qu, J. Wu, Y. Zhang, L. Jia, Y. Yang, X. Xu, S. T. Chu, B. E. Little, R. Morandotti, B. Jia, and D. J. Moss, "Graphene oxide for enhanced optical nonlinear performance in CMOS compatible integrated devices", Paper No. 11688-30, PW21O-OE109-36, 2D Photonic Materials and Devices IV, SPIE Photonics West, San Francisco CA March 6-11 (2021). doi.org/10.1117/12.2583978

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119. Yang Qu, Jiayang Wu, Yunyi Yang, Yuning Zhang, Yao Liang, Houssein El Dirani, Romain Crochemore, Pierre Demongodin, Corrado Sciancalepore, Christian Grillet, Christelle Monat, Baohua Jia, and David J. Moss, "Enhanced nonlinear four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films", Advanced Optical Materials vol. 8 (21) 2001048 (2020). DOI: 10.1002/adom.202001048. arXiv:2006.14944.

120. Yuning Zhang, Yang Qu, Jiayang Wu, Linnan Jia, Yunyi Yang, Xingyuan Xu, Baohua Jia, and David J. Moss, "Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films", ACS Applied Materials and Interfaces vol. 12 (29) 33094-33103 June 29 (2020). DOI:10.1021/acsami.0c07852

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121. Jiayang Wu, Yunyi Yang, Yang Qu, Yuning Zhang, Linnan Jia, Xingyuan Xu, Sai T. Chu, Brent E. Little, Roberto Morandotti, Baohua Jia, and David J. Moss, "Enhanced nonlinear four-wave mixing in microring resonators integrated with layered graphene oxide films", Small vol. 16 (16) 1906563 (2020). DOI: 10.1002/smll.201906563

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122. Jiayang Wu, Yunyi Yang, Yang Qu, Xingyuan Xu, Yao Liang, Sai T. Chu, Brent E. Little, Roberto Morandotti, Baohua Jia, and David J. Moss, "Graphene oxide waveguide polarizers and polarization selective micro-ring resonators", Paper 11282-29, SPIE Photonics West, San Francisco, CA, 4 - 7 February (2020). doi: 10.1117/12.2544584

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123. Jiayang Wu, Yunyi Yang, Yang Qu, Xingyuan Xu, Yao Liang, Sai T. Chu, Brent E. Little, Roberto Morandotti, Baohua Jia, and David J. Moss, "Graphene oxide waveguide polarizers and polarization selective micro-ring resonators", Laser and Photonics Reviews vol. 13 (9) 1900056 (2019). DOI:10.1002/lpor.201900056.

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124. Yunyi Yang, Jiayang Wu, Xingyuan Xu, Sai T. Chu, Brent E. Little, Roberto Morandotti, Baohua Jia, and David J. Moss, "Enhanced four-wave mixing in graphene oxide coated waveguides", Applied Physics Letters Photonics vol. 3 120803 (2018). doi: 10.1063/1.5045509.

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125. Linnan Jia, Yang Qu, Jiayang Wu, Yuning Zhang, Yunyi Yang, Baohua Jia, and David J. Moss, "Third-order optical nonlinearities of 2D materials at telecommunications wavelengths", Micromachines (MDPI), 14, 307 (2023). https://doi.org/10.3390/mi14020307.

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126. Linnan Jia, Dandan Cui, Jiayang Wu, Haifeng Feng, Tieshan Yang, Yunyi Yang, Yi Du, Weichang Hao, Baohua Jia, David J. Moss, "BiOBr nanoflakes with strong nonlinear optical properties towards hybrid integrated photonic devices", Applied Physics Letters Photonics vol. 4 090802 vol. (2019). DOI: 10.1063/1.5116621

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127. D. Jin, W. Liu, L. Jia, Y. Zhang, J. Hu, H. El Dirani, S. Kerdiles, C. Sciancalepore, P. Demongodin, C. Grillet, C. Monat, D. Huang, J. Wu, B. Jia, and D. J. Moss, "Thickness- and wavelength-dependent nonlinear optical absorption in 2D layered MXene films," *Small Science*, vol. n/a, no. n/a, pp. 2400179.

128. Linnan Jia, Jiayang Wu, Yunyi Yang, Yi Du, Baohua Jia, David J. Moss, "Large Third-Order Optical Kerr Nonlinearity in Nanometer-Thick PdSe2 2D Dichalcogenide Films: Implications for Nonlinear Photonic Devices", ACS Applied Nano Materials vol. 3 (7) 6876-6883 (2020). DOI:10.1021/acsnanm.0c01239.

129. H. Yu, S. Sciara, M. Chemnitz, N. Montaut, B. Fischer, R. Helsten, B. Crockett, B. Wetzel, T. A. Göbel, R. Krämer, B. E. Little, S. T. Chu, D. J. Moss, S. Nolte, W.J. Munro, Z. Wang, J. Azaña, R. Morandotti, "Quantum key distribution implemented with d-level time-bin entangled photons", Nature Communications Vol. 16, 171 (2025). DOI:10.1038/s41467-024-55345-0.

130. Weiwei Han, Zhihui Liu, Yifu Xu, Mengxi Tan, Yuhua Li, Xiaotian Zhu, Yanni Ou, Feifei Yin, Roberto Morandotti, Brent E. Little, Sai Tak Chu, David J. Moss, Xingyuan Xu, and Kun Xu, "TOPS-speed complex-valued convolutional accelerator for feature extraction and inference", Nature Communications Vol. 16, 292 (2025). DOI: 10.1038/s41467-024-55321-8.

131. Yonghang Sun, James Salamy, Caitlin E. Murray, Xiaotian Zhu, Brent E. Little, Roberto Morandotti, Arnan Mitchell, Sai T. Chu, David J. Moss, Bill Corcoran, "Self-locking of free-running DFB lasers to a single microring resonator for dense WDM", Journal of Lightwave Technology, (2024). DOI: 10.1109/JLT.2024.3494694.

132. Zhihui Liu, Haoran Zhang, Yuhang Song, Xiaotian Zhu, Yunping Bai, Mengxi Tan, Bill Corcoran, Caitlin Murphy, Sai T. Chu, David J. Moss, Xingyuan Xu, and Kun Xu, "Advances in Soliton Crystals Microcombs", Photonics Vol. 11, 1164 (2024). https://doi.org/10.3390/photonics11121164.

133. Di Jin, Jiayang Wu, Sian Ren, Junkai Hu, Duan Huang, and David J. Moss, "Modeling of complex integrated photonic resonators using scattering matrix method", Photonics Vol. 11, 1107 (2024). https://doi.org/10.3390/photonics11121107.

134. Yonghang Sun, James Salamy, Caitlin E. Murry, Brent E. Little, Sai T. Chu, Roberto Morandotti, Arnan Mitchell, David J. Moss, Bill Corcoran, "Enhancing laser temperature stability by passive self-injection locking to a microring resonator", Optics Express Vol. 32 (13) 23841-23855 (2024). https://doi.org/10.1364/OE.515269.

135. C. Khallouf, V. T. Hoang, G. Fanjoux, B. Little, S. T. Chu, D. J. Moss, R. Morandotti, J. M. Dudley, B. Wetzel, and T. Sylvestre, "Raman scattering and supercontinuum generation in high-index doped silica chip waveguides", Nonlinear Optics and its Applications, edited by John M. Dudley, Anna C. Peacock, Birgit Stiller, Giovanna Tissoni, SPIE Vol. 13004, 130040I (2024). doi: 10.1117/12.3021965

136. Yang Li, Yang Sun, Jiayang Wu, Guanghui Ren, Roberto Morandotti, Xingyuan Xu, Mengxi Tan, Arnan Mitchell, and David J. Moss, "Performance analysis of microwave photonic spectral filters based on optical microcombs", Advanced Physics Research Vol. 3 (9) 2400084 (2024). DOI:10.1002/apxr.202400084.

137. Di Jin, Jiayang Wu, Junkai Hu, Wenbo Liu1, Yuning Zhang, Yunyi Yang, Linnan Jia, Duan Huang, Baohua Jia, and David J. Moss, "Silicon photonic waveguide and microring resonator polarizers incorporating 2D graphene oxide films", Applied Physics Letters Vol. 125, 053101 (2024). doi: 10.1063/5.0221793.

138. Jiayang Wu, Yuning Zhang, Junkai Hu, Yunyi Yang, Di Jin, Wenbo Liu, Duan Huang, Baohua Jia, David J. Moss, "Novel functionality with 2D graphene oxide films integrated on silicon photonic chips", Advanced Materials Vol. 36 2403659 (2024). DOI: 10.1002/adma.202403659.

139. Yuning Zhang, Jiayang Wu, Linnan Jia, Di Jin, Baohua Jia, Xiaoyong Hu, David Moss, Qihuang Gong, "Advanced optical polarizers based on 2D materials", npj Nanophotonics 1 (2024). DOI: 10.1038/s44310-024-00028-3.
140. Junkai Hu, Jiayang Wu, Wenbo Liu, Di Jin, Houssein El Dirani, Sébastien Kerdiles, Corrado Sciancalepore, Pierre Demongodin, Christian Grillet, Christelle Monat, Duan Huang, Baohua Jia, and David J. Moss, "2D graphene oxide: a versatile thermo-optic material", Advanced Functional Materials Vol. 34 2406799 (2024). DOI: 10.1002/adfm.202406799.

141. Di Jin, Wenbo Liu, Linnan Jia, Junkai Hu, Duan Huang, Jiayang Wu, Baohua Jia, and David J. Moss, "Thickness and Wavelength Dependent Nonlinear Optical Absorption in 2D Layered MXene Films", Small Science Vol. 4 2400179 (2024). DOI:10.1002/smsc202400179.

142. Andrew Cooper, Luana Olivieri, Antonio Cutrona, Debayan Das, Luke Peters, Sai Tak Chu, Brent Little, Roberto Morandotti, David J Moss, Marco Peccianti, and Alessia Pasquazi, "Parametric interaction of laser cavity-solitons with an external CW pump", Optics Express Vol. 32 (12), 21783-21794 (2024).

143. Weiwei Han, Zhihui Liu, Yifu Xu, Mengxi Tan, Chaoran Huang, Jiayang Wu, Kun Xu, David J. Moss, and Xingyuan Xu, "Photonic RF Channelization Based on Microcombs", Special Issue on Microcombs IEEE Journal of Selected Topics in Quantum Electronics Vol. 30 (5) 7600417 (2024). DOI:10.1109/JSTQE.2024.3398419.

144. Y. Li, Y. Sun, J. Wu, G. Ren, X. Xu, M. Tan, S. Chu, B. Little, R. Morandotti, A. Mitchell, and D. J. Moss, "Feedback control in micro-comb-based microwave photonic transversal filter systems", IEEE Journal of Selected Topics in Quantum Electronics Vol. 30 (5) 2900117 (2024). DOI: 10.1109/JSTQE.2024.3377249.

145. Weiwei Han, Zhihui Liu, Yifu Xu, Mengxi Tan, Yuhua Li, Xiaotian Zhu, Yanni Ou, Feifei Yin, Roberto Morandotti, Brent E. Little, Sai Tak Chu, Xingyuan Xu, David J. Moss, and Kun Xu, "Dual-polarization RF Channelizer Based on Microcombs", Optics Express 32, No. 7, 11281-11295 / 25 Mar 2024 / (2024). DOI: 10.1364/OE.519235.

146. Aadhi A. Rahim, Imtiaz Alamgir, Luigi Di Lauro, Bennet Fischer, Nicolas Perron, Pavel Dmitriev, Celine Mazoukh, Piotr Roztocki, Cristina Rimoldi, Mario Chemnitz, Armaghan Eshaghi, Evgeny A. Viktorov, Anton V. Kovalev, Brent E. Little, Sai T. Chu, David J. Moss, and Roberto Morandotti, "Mode-locked laser with multiple timescales in a microresonator-based nested cavity", APL Photonics Vol. 9 031302 (2024). DOI:10.1063/5.0174697.

147. C. Mazoukh, L. Di Lauro, I. Alamgir, B. Fischer, A. Aadhi, A. Eshaghi, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, "Genetic algorithm-enhanced microcomb state generation", Special Issue Microresonator Frequency Combs - New Horizons, Nature Communications Physics Vol. 7, Article: 81 (2024) (2024). DOI: 10.1038/s42005-024-01558-0.

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