Flexibly Reconfigurable Kerr Micro-Comb Based on Cascaded Si₃N₄ Micro-Ring Filters

Abstract

In recent years, micro-combs, due to their compact structure and high efficiency, have proven to be a practical solution for optical sources. In this paper, an approach to flexibly modulating micro-combs is proposed, and a simulation platform based on ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-combs with highly integrated, tunable, and reconfigurable features is built. By means of the Lugiato–Lefever equation model, the dynamic evolution process of micro-combs is analyzed, and a micro-ring resonator is designed with a free spectral range of 7.24 nm, an effective mode area of $1.0829{\mu \mathrm{m}}^2$, and coherent comb lines spanning over 125 THz. Cascaded silicon nitride micro-ring filters are utilized to obtain reconfigurable modulation effects for Kerr-frequency micro-combs. Due to the significance of flexibly controlled optical sources with high-repetition rates and multiple channels for system-on-chip, our proposal has potential in photonic integrated circuit systems, such as high-density photonic computing and large-capacity optical communications, in the future.

1. Introduction

Thanks to equal spacing of comb lines and their coherence, optical frequency combs served as broadband optical sources possess extensive applications in fields such as precision measurement [1], microwave photonics [2], quantum optics [3], optical communications [4,5], photonic computing [6,7], and so on. Particularly in photonic computing, these combs act as multi-wavelength coherent optical sources that enhance the computational capabilities of convolutional neural networks [8]. Given the demands for high repetition rates, large capacities, and multi-channel capabilities in photonic computing, leveraging the nonlinear effects of optical micro-ring resonators alongside the technological advantages of silicon-based optoelectronic components is essential. This approach aims to develop highly integrated, stable, and coherent micro-cavity optical frequency comb systems, thereby significantly improving high-density computing and high-speed optical communication capabilities.

Optical frequency combs can be obtained using various devices, including mode-locked lasers [9], electro-optic modulators [10], and highly nonlinear optical fibers [11]. Although these approaches have advantages such as a broad output spectrum, low system complexity, and low costs, the micro-cavity-based scheme offers superior benefits, such as smaller component sizes, higher integration, higher stability, and improved coherence [12,13,14]. In addition, this proposal also enables the control and tuning of more comb lines and different frequency intervals. The tuning of micro-combs can be classified into several categories: ‘power kicking’, frequency tuning, thermal tuning, and pump phase modulation. Fundamentally, these approaches adjust the detuning, which is the difference between the wavelength of the pump light and that of the micro-cavity resonance, to trigger the nonlinear Kerr effect and generate the optical frequency comb.

Moreover, the generation of a micro-comb depends on the nonlinear Kerr effect, and a higher nonlinear coefficient of the micro-cavity material results in a lower power threshold for triggering the four-wave mixing effect, thereby facilitating the excitation of the frequency comb [15]. Silicon nitride (${\mathrm{Si}}_3{\mathrm{N}}_4$), known for its broad transparency range (400–4000 nm) and extremely low optical transmission loss (0.05–0.2 dB/cm) [16], allows components to achieve a high quality factor (referred to as the Q-factor). Compared to materials such as Si and AlGaAs, ${\mathrm{Si}}_3{\mathrm{N}}_4$ has a moderate refractive index (around 2), which prevents issues such as excessively large effective mode areas caused by a low refractive index. Additionally, ${\mathrm{Si}}_3{\mathrm{N}}_4$ materials exhibit excellent thermal conductivity [17], making the thermal tuning of the comb more feasible. Compared with other tuning methods, ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-comb generation schemes based on frequency tuning cover comparatively larger spectral ranges, including the C-band [18].

In 2010, Michal Lipson used existing integration processes to fabricate integrated waveguide micro-ring resonators, obtaining the first optical frequency comb spectrum based on ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-cavities, with over 100 spectral lines and a spacing of hundreds of GHz [19]. In 2017, H. Guo and T. J. Kippenberg et al. achieved the maintenance of a single soliton state in a ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-cavity by combining forward and backward frequency tuning, observing unique double-resonance characteristics in the soliton state [20]. In the same year, Pfeiffer et al. obtained broadband soliton optical frequency combs with octave-spanning spectra by designing the high-order dispersion of a ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-cavity [21]. In 2019, B. Y. Kim obtained dark solitons with a conversion efficiency of up to 41% based on an on-chip normal-dispersion ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-cavity [22]. Also, in 2019, H. Zhou obtained a stable single soliton state using auxiliary light heating methods based on on-chip ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-rings [23]. In 2024, T. Jiang proposed a mode-division multiplexing scheme, using a single laser to pump adjacent fundamental and high-order modes of a single integrated ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-ring resonator, realizing coherent dual-combs with a repetition rate of 50 GHz [24].

Although the spectrum space of optical frequency combs and the deterministic generation of solitons have been deeply investigated, flexibly tuning the number and spacing of comb lines based on a monolithic chip is still to be further explored. This proposed scheme is based on the nonlinear Kerr effect, activated via a ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-cavity with a comparatively small mode area, as well as the thermo-optic effects and transmission characteristics of a cascaded micro-ring resonator. Compared to another ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-ring with an effective mode area of $1.15{\mu \mathrm{m}}^2$ [25] and power conversion efficiency of 23% [26], our designed micro-resonator possesses a smaller effective mode area of $1.0829{\mu \mathrm{m}}^2$, and our micro-comb demonstrates an improved conversion efficiency of 30%. The enhancement in power conversion efficiency is able to confer superior performance to silicon photonic systems. The dynamic evolution of the optical frequency comb is observed in both the time and frequency domains, and combs based on micro-ring resonators with 1/2, 1/3, and 1/4 multiples of radii have a free spectral range (FSR) of around 14 nm, 21 nm, and 28 nm, respectively, which demonstrates the flexible selectivity and free tunability of comb line positions. This designed component can be employed as a high-repetition-rate and multi-channel optical source to address the limitations of traditional optical frequency comb generation schemes with poor integration and high power consumption and satisfy the requirements for high-speed paralleled transmission and high spectral efficiency in photonic computing and optical communications.

2. Component Design

To meet the phase-matching condition required to trigger the four-wave mixing effect during the generation of micro-combs, the micro-cavity must exhibit anomalous dispersion [27]. A thicker waveguide is advantageous as it can provide lower losses and better confine the optical field, resulting in a higher intensity within the waveguide. This strengthened optical intensity improves the nonlinear effects within the micro-cavity, facilitating the satisfaction of the anomalous dispersion condition. Based on the above analysis, due to the high damage threshold and low transmission loss, 800 nm is chosen as the ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide’s thickness. Furthermore, silicon nitride with an 800 nm thickness offers flexibility in dispersion engineering and tuning, high optical mode confinement, and a low power threshold for exciting nonlinear optics effects. In addition, with the help of the Finite-Difference Eigenmode (FDE) solver, the cross-section of the 800 nm thick waveguide is discretized into grids, and Maxwell’s equations are solved at each grid point to calculate the spatial distribution of modes. This approach allows us to determinate various parameters of the waveguide structure, including the mode field distribution, effective refractive index, loss, and dispersion. We simulate the dispersion curves for waveguides with different widths at a thickness of 800 nm. After the comparison and analysis of the flatness of these dispersion curves, a width of 1500 nm for the ${\mathrm{Si}}_3{\mathrm{N}}_4$ waveguide is selected, as depicted in Figure 1a, guaranteeing that the micro-cavity achieves anomalous dispersion near the C-band. The inset in Figure 1a shows that the strong mode distribution is in the center of waveguide’s cross-section with bright color at the wavelength of 1550 nm. Compared with the 800 nm-spanning comb generated by a ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-ring with a waveguide thickness of 740 nm and width of 1500 nm [28], the ${\mathrm{Si}}_3{\mathrm{N}}_4$ micro-ring with a thickness of 800 nm and width of 1500 nm theoretically offers broader spectral coverage due to the increased thickness, which will be further elaborated upon in the subsequent sections of this paper. This improvement arises from the compression of the optical field distribution, the reduction in mode volume, and the alteration of the dispersion curve slope.

Simultaneously, the effective refractive indices of waveguides with different widths at a thickness of 800 nm are shown in Figure 1b. Here, the red and green lines represent the first and second TE-like modes, respectively, and the dark and blue lines show the first and second TM-like modes, respectively. As illustrated in Figure 1b, at around an 800 nm width, the first TE and first TM modes cross. To guarantee a single-mode operation status and obtain total internal reflection, the effective refractive index of the core waveguide needs to be higher than that of the silica cladding (1.44), which is indicated by a dashed line. In Figure 1b, it is obvious that when the core waveguide width is around 700 nm, the waveguide supports only a single TE fundamental mode and a single TM fundamental mode, thus satisfying the single-mode condition. Therefore, to satisfy the single-mode condition requirement, the bus waveguide width in the micro-cavity is selected to be $0.7\mu \mathrm{m}$, and the ring waveguide width is designed as $1.5\mu \mathrm{m}$ to obtain anomalous dispersion.

Additionally, as illustrated in Figure 2a, a bent coupling approach is employed to increase the area and duration of coupling from the bus waveguide to the ring, which shows the designed micro-ring resonator’s structure for optical frequency comb generation. The central angle corresponding to the bent waveguide is 90 degrees. By observing the field distribution of microcavities that contain straight and bent waveguides in Figure 2b,c, it can be concluded that the structure designed by us can increase the coupling of light in the ring and enable better resonance of light within the ring. Using the aforementioned simulation methods, the effective refractive index, $n_{\mathrm{eff}}$, of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ intra-ring cavity is approximately $1.8076$, and the effective mode area, $A_{\mathrm{eff}}\approx 1.0829\times {10}^{12}{\mathrm{m}}^2$, and second-order dispersion coefficient, ${\beta }_2\approx -7.785\times {10}^{-26}{\mathrm{s}}^2/\mathrm{m}$, are obtained, respectively, with a pump power of $P_{\mathrm{in}}=1\mathrm{W}$. The radius, R, of the micro-ring is set as $25\mathsf{\mu }\mathrm{m}$; thus, the effective mode volume, $V_{\mathrm{eff}}\approx 1.701\times {10}^{-16}{\mathrm{m}}^3$, can be calculated from $V_{\mathrm{eff}}=A_{\mathrm{eff}}L$. The nonlinear refractive index coefficient, $n_2$, for the ${\mathrm{Si}}_3{\mathrm{N}}_4$ material is approximately $2.4\times {10}^{-19}{\mathrm{m}}^2/\mathrm{W}$ [29], which can be used to determine the nonlinear coefficients $\gamma ={\omega }_0{n}_2/c{A}_{\mathrm{eff}}\approx 0.8984{\mathrm{m}}^{-1}{\mathrm{W}}^{-1}$ and ${\omega }_0=2\pi c/{\lambda }_0$, where ${\lambda }_0=1550\mathrm{nm}$.

The waveguide structures are designed to obtain the transmission spectra of the micro-cavity through simulation. By observing the number of transmission peaks within a single FSR, we analyze the number of excited modes. Figure 2d illustrates the transmission spectrum at the drop port of the micro-ring resonator, with a structure of 1500 nm × 800 nm for the ring waveguide and 700 nm × 800 nm for the bus waveguide, obtaining an FSR of approximately 7 nm, which satisfies the requirements for the photonic computing of light sources. In Figure 2d, there is only one transmission peak within a single FSR, indicating the excitation of a single fundamental mode. As the transmission spectra for both the TE and TM fundamental modes are similar in the numerical simulation, it is important to emphasize that all the transmission spectra presented in this work are based on the TE mode. And also, Figure 2e comes from the blue shaded area of Figure 2d, which shows that the Q-factor of the ring resonator can reach approximately 2.6 × ${10}^4$. If the coupling distance between the ring and bus waveguides increases, more transmission peaks appear within a single FSR, indicating the excitation of multiple modes. This configuration can achieve a Q-factor of beyond ${10}^6$, which strengthens the nonlinear effects and reduces the threshold power for optical frequency comb generation and the intra-cavity loss, as well as improving frequency stability. However, even if the Q-factor of the micro-cavity does not reach an extremely high level, stable optical frequency combs can still be obtained by pump lasers with a higher power, low-loss materials, and optimized fabrication processes. Therefore, one single-mode structure is employed, refining the coupling gap to closer to the critical coupling status for the micro-ring. Thanks to the high nonlinearity and low loss characteristics of ${\mathrm{Si}}_3{\mathrm{N}}_4$ as the waveguide material, subsequent simulations for micro-comb generation are implemented with a comparatively high pump power.

3. Numerical Simulation

Numerical simulation is carried using mathematical modeling of the Lugiato–Lefever equation (LLE), which is a combination of the nonlinear Schrödinger equation and the boundary conditions for optical field transmission in the cavity. The split-step Fourier method is applied to handle the LLE, solving the nonlinear term in the time domain and the dispersion term in the frequency domain. The LLE model offers the advantage of directly solving the optical field in the time domain, necessitating significantly less computation time compared to the coupled-mode equation that describes the dynamics of micro-combs in the frequency domain [30]. The mathematical model for the transmission of the optical field in the cavity is derived from the LLE, which is formulated as follows [31]:

$$t_R{\displaystyle \frac{\partial E(t,\tau )}{\partial t}}=\left[-{\displaystyle \frac{{\alpha }_i+\theta }{2}}-i{\delta }_0+iL\sum _{k\ge 2}{\displaystyle \frac{{\beta }_k}{k!}}{\left(i{\displaystyle \frac{\partial }{\partial \tau }}\right)}^k+iL\gamma {\left|E\right|}^2\right]E+\sqrt{\theta }{E}_{\mathrm{in}}.$$

(1)

Here, $t_R$ represents the round-trip time for light in the cavity, and E denotes the slowly varying amplitude of the intra-cavity optical field in the time domain. In addition, t, known as slow time, indicates the evolution of the timescale over multiple round trips. In contrast, $\tau$, also referred to as fast time, characterizes the time coordinate within a single round-trip timescale. $\alpha =({\alpha }_i+\theta )/2$ represents the intra-cavity total loss, where ${\alpha }_i$ is the linear transmission loss, and $\theta$ is the coupling coefficient. L denotes the perimeter of the micro-ring resonator, and ${\delta }_0=({\omega }_0-{\omega }_p)t_R$, where ${\omega }_0$ and ${\omega }_p$ represent the resonant angular frequency of the micro-cavity and the central angular frequency of the pump light, respectively. When ${\omega }_0-{\omega }_p<0$, the system is in a blue-detuned state, while ${\omega }_0-{\omega }_p>0$ means it is in a red-detuned state. In this work, we paid more attention to second-order dispersion, which plays a dominant role in the formation and stability of the optical frequency comb, while neglecting higher-order dispersions with minor impacts.

We set the frequency modes to $2^9$ to strengthen the numerical simulation efficiency. Initially, as shown in Figure 3a, the pump’s initial detuning was set to −0.005, being kept in the blue-detuned region, and Gaussian noise was added to the pump to mimic experimental disturbances. Additionally, its frequency domain corresponds to a single comb line at a wavelength of 1550 nm in Figure 3b. The central wavelength of the pump light was then adjusted, being swept from the blue-detuned to the red-detuned region at a rate of 2 × ${10}^{-3}$/ns. The tuning was halted once a comparatively stable multi-soliton state was achieved in the cavity. Figure 3 illustrates that both intra-cavity optical solitons and Kerr optical frequency combs sequentially undergo the main stages of Turing rolls, chaos, multi-soliton phenomena, and single-soliton phenomena, respectively.

4. Results and Discussion

When the pump light is in the blue-detuned region (region I in Figure 4a), the sweeping frequency makes the optical power increase in the cavity, which gradually reaches the threshold power required to trigger the Kerr nonlinear effect. Due to degenerate four-wave mixing, the main comb is centered around a 1550 nm wavelength for the intracavity spectrum. When the pump light approaches the resonance peak, higher-order sidebands are generated by cascaded four-wave mixing between the pump light and the main combs, as well as between the combs themselves. This stage, known as the Turing roll state, is depicted in Figure 3c,d for the time and frequency domains, respectively. It appears as periodic pulses in the time domain and is shown as low-noise main combs with coherence in the frequency domain.

When the wavelength of the pump laser sweeps over the resonance peak and enters the red-detuned region, the state of the micro-cavity transitions to a chaotic state due to modulation instability. As illustrated in region II of Figure 4a, by further adjusting the pump detuning from the Turing roll state, the intracavity power rapidly increases and shows irregular fluctuations. The optical comb loses its stability, and the amplitude of each frequency mode fluctuates significantly, resulting in high-noise and incoherent optical combs. As shown in Figure 3e,f, the time domain waveform reveals unstable and random jittering.

As depicted in Figure 3g,h, by further detuning, the intracavity power diminishes, leading to the emergence of multiple stable pulses in the time domain, which indicates a multi-soliton state. As the soliton state emerges from the chaotic state, the number of generated solitons is random. During the transition from the chaotic to the multi-soliton state, there is a change from multi-pulse to single-pulse behavior, as indicated by the brown circled area in Figure 4b (at this point, the stable multi-soliton state has not been fully established yet). The evolution of intracavity power in region III of Figure 4a shows a stepwise decrease in power. This occurs because the power drops rapidly over a short period, which reduces the parametric gain of soliton pulses. When the gain is lower than the loss, multiple solitons decay due to intracavity losses. During this process, adjacent solitons can be merged into an individual soliton, which brings about “large steps” that represent the simultaneous decay of multiple solitons observed in the power evolution of Figure 4a, yet “small steps” indicate the annihilation of individual solitons, as depicted in region IV of Figure 4a. At this stage, the optical comb has already occupied each resonance mode in the frequency domain.

Along with the decrease in intracavity power and the annihilation of solitons, balance is obtained between dispersion and nonlinearity, as well as between gain and loss. As shown in Figure 3i,j, the intracavity power stabilizes, bringing about the presence of single stable pulse in the time domain, which indicates a single-soliton state. In the frequency domain, this state reveals a smooth envelope with a ${\mathrm{sech}}^2$ shape, indicating a mode-locked state. The pump-to-comb conversion efficiency [32] can be calculated as the ratio of the power of the output comb lines to the input pump power, excluding the pump frequency, which is approximately 30% and able to be increased by adjusting parameters such as the dispersion coefficient, the coupling coefficient, and the FSR of the resonator [33]. This single-soliton state contributes to broadband optical frequency combs spanning over 125 THz (from 1200 nm to 2200 nm) with a repetition rate of approximately 906 GHz. Furthermore, the comb lines maintain stable phase relations, guaranteeing high coherence. It is noteworthy that solitons only exist in the effective red-detuned region.

We leverage the transmission characteristics of the micro-ring resonator to obtain a filtering effect. As illustrated from the system’s schematic diagram in Figure 5, based on the spacing of the optical comb lines (namely, the FSR of the micro-cavity), the structural parameters of the filters are designed such that several cascaded micro-ring filters are configured with a double FSR, a triple FSR, and a quadruple FSR for comb generation. The optical signal from the tunable laser diode is amplified by a high-power, erbium-doped fiber amplifier (EDFA) to serve as the pump light. After passing through a polarization controller (PC) to adjust the polarization state, the light is coupled in the micro-ring resonator. When the pump light wavelength matches the resonant peak of the resonator, a resonant enhancement effect occurs, facilitating four-wave mixing and bringing about a high-intensity optical field in the cavity. By selecting an appropriate power level and adjusting the sweeping steps of the tunable continuous-wave laser, Kerr nonlinearity is introduced to generate combs with the micro-comb resonator. Moreover, a model is built for the silicon nitride micro-ring resonator, and Au electrodes and a TiN resistor structure are constructed in Ansys Lumerical Heat Solutions. The thermal and electrical boundary conditions are configured as follows. A constant temperature of 300 K is defined at the thermal boundary on the surface of the Au electrodes and silicon substrate to support steady-state and transient modes. The electrical boundary condition is also utilized in thermal and conductive simulations to apply a bias voltage to the electrical contacts of the simulated structure. Additionally, the convection effect between the metal and the ambient air is taken into account to obtain better heat transfer. To enhance the authenticity and precision of the simulation, the overall meshes are placed in the Au and TiN regions, and the mesh constraints are configured as a max edge length of $0.1\mu \mathrm{m}$ for all the meshes. Along with TiN resistors and Au electrodes covered on the surface of the cascaded micro-ring waveguides to enable thermal control, cascaded micro-ring filters are then connected. By means of the thermo-optic effect, the effective refractive index of the material can be altered. After adjusting the resonant wavelength of the micro-cavity filters, flexible and adjustable optical frequency combs with a comb line spacing equal to integer multiples of the FSR can be obtained. Finally, the characteristics of the optical frequency combs can be monitored using an optical spectrum analyzer (OSA) and oscilloscope. Above all, both the cascaded micro-ring filters and the micro-comb resonator are made of ${\mathrm{Si}}_3{\mathrm{N}}_4$ material, integrated onto the same platform and connected through a Y-branch waveguide.

A voltage ranging from 0 to 4 V (with intervals of 0.5 V) is applied to the Au electrodes for heat conduction. The transmission spectra at the drop port for the triple add-drop micro-ring filters applied with variable voltages are shown in Figure 6, and the wavelengths of the resonant peaks are marked by red circles. As the applied voltages increase, the offsets of the resonant peak red-shift increase accordingly for ${\mathrm{MRR}}_1$, ${\mathrm{MRR}}_2$, and ${\mathrm{MRR}}_3$ because of thermo-optic effects. In addition, the drifting of the resonant wavelength rises as the FSR increases with an equal voltage applied, and the 3-dB bandwidth of each micro-ring filter becomes broader as well. Moreover, the structural parameters are optimized to guarantee that the 3-dB bandwidth of the triple-ring is relatively less than the respective comb spacing, thus achieving fine filtering effects. As shown in Figure 7a, the relations between the intra-cavity temperature and the applied voltage on triple-ring waveguides are analyzed using the finite element method (FEM). Obviously, the heating temperature is positively correlated with the applied voltages. As depicted in Figure 7b–d, the resonant peaks of the filters are finely adjusted to align with the wavelengths of the optical frequency combs, and the 3-dB bandwidths of ${\mathrm{MRR}}_1$, ${\mathrm{MRR}}_2$, and ${\mathrm{MRR}}_3$ at 1550 nm are 2.64 nm, 3.45 nm, and 4.2 nm, respectively. At the same time, the extinction ratios of the triple-ring drop as the radii of the corresponding micro-ring decrease, which is illustrated in Figure 7b–d. Therefore, an EDFA is subsequently distributed at the drop port of ${\mathrm{MRR}}_3$ to improve its filtering performance for higher extinction ratios. In consideration of possible coupling effects and potential interference between cascaded micro-rings, which may bring about extra transmission loss, a sufficient spatial distance is reserved between each micro-cavity to ensure that the triple-ring filters do not interfere with each other.

A pump light source with an output power of 1 W (30 dBm), as depicted in the system in Figure 5, is utilized as the CW tunable laser. After passing through the Y-branch waveguide at point A, the output optical signal of the micro-comb resonator is split into two parts as the input optical signals of ${\mathrm{MRR}}_1$ and ${\mathrm{MRR}}_2$, whose optical powers are both 16.76 dBm. Similarly, the output light of ${\mathrm{MRR}}_1$ at its through port becomes the input optical signal of ${\mathrm{MRR}}_3$, which is 15.86 dBm. Based on the above results, the wavelength responses of ${\mathrm{MRR}}_1$, ${\mathrm{MRR}}_2$, and ${\mathrm{MRR}}_3$ are obtained, respectively, as depicted in Figure 8. Furthermore, as illustrated in Figure 8, the extinction ratios (ERs), denoted as $ER=10lg\left(P_{max}/P_{min}\right)$, of ${\mathrm{MRR}}_1$, ${\mathrm{MRR}}_2$, and ${\mathrm{MRR}}_3$ at a wavelength of 1550 nm are 11.81 dB, 12.91 dB, and 13.76 dB, respectively, which can be obtained from the transmission power ratio of the resonant peak to valley at the drop port. In addition, the ER can also be derived from the ratio of the maximum to the minimum power of the normalized transmission spectra at a specific wavelength, as shown in Figure 7b–d.

Insertion loss (IL), expressed as $IL=-10lg\left(T_{max}\right)$, can be calculated from the ratio of the peak normalized transmission (the amplitude in Figure 7b–d) to the ideal transmission, and the normalized value of “ideal transmission” is 1. Thus, the insertion losses of ${\mathrm{MRR}}_1$, ${\mathrm{MRR}}_2$, and ${\mathrm{MRR}}_3$ at 1550 nm can be obtained from our simulation results, which are 0.47 dB, 1.1 dB, and 2.64 dB, respectively. A lower insertion loss means that less energy will be lost during the optical signal transmission if the component structure is designed with superior optical field confinement. Notably, the IL can be optimized by selecting materials with lower absorption coefficients or precisely controlling the parameters of the micro-ring resonator to avoid mode mismatch and scattering.

Moreover, the required frequency components for photonic computing and the number of comb lines can be selected, and the relevant parameters can be configured, such as the corresponding wavelength and power of the comb lines in the system-level simulation. As shown in Figure 9a, nine comb lines in the C-band ranging from 1520 nm to 1582 nm are selected (at point A in Figure 5). Simultaneously, in Figure 9b–d, it is demonstrated that the tunable micro-comb at the drop ports of ${\mathrm{MRR}}_1$, ${\mathrm{MRR}}_2$, and ${\mathrm{MRR}}_3$ (points B, C, and D in Figure 5) are all centered around 1550 nm, with the comb line spacing of the double FSR, the triple FSR, and the quadruple FSR, respectively, which allows for autonomous adjusting for comb line spacing. Furthermore, the parameter of finesse, defined as $F=\left(FSR/FWHM\right)$, is widely used to assess the filter performance. Herein, the full width at half maximum (FWHM) is equal to a 3-dB bandwidth. Therefore, according to the corresponding FSRs and FWHMs, the finesses of our ${\mathrm{MRR}}_1$, ${\mathrm{MRR}}_2$, and ${\mathrm{MRR}}_3$ are calculated as 5.36, 6.19, and 6.81, respectively. In addition, for multi-band micro-ring filters, the out-of-band rejection ratio (OBRR) is a key metric for evaluating the filtering effects. Actually, the OBRR can be obtained through the analysis of the output optical spectra of micro-comb and cascaded micro-ring filters, as illustrated in Figure 9. By comparing between the output optical spectra before and after being filtered, the OBRRs of ${\mathrm{MRR}}_1$ and ${\mathrm{MRR}}_2$ are obtained, which are approximately 14.28 dB and 15.29 dB, respectively, at a wavelength of 1542.7 nm. Obviously, our cascaded micro-ring filters improve the filtering efficiency thanks to multiple passbands and, at the same time, achieve a desirable suppression effect for the signals’ spectral components out of the passband, which brings flexible control to the output signal of the optical frequency comb.

5. Conclusions

In this paper, we build a highly integrated, flexible, and tunable on-chip ${\mathrm{Si}}_3{\mathrm{N}}_4$ platform for optical frequency comb generation based on theoretical analysis and numerical simulations. By employing the LLE model, the dynamic evolution of the optical comb is analyzed. Forward frequency tuning facilitates the generation of optical comb states, transitioning from Turing patterns to chaos, multi-soliton, and single-soliton stages in turn. The coherent optical combs are created with a temporal single soliton and a remarkably smooth envelope, featuring a spectral span of 125 THz and a repetition rate of approximately 906 GHz. During the tuning process, step-like variations in power are observed in the cavity, with instances of multiple solitons merging into a single soliton or annihilating directly, leading to a decrease with a “large step” in power.

Subsequently, we leverage the filtering characteristics of a cascaded micro-ring resonator and the thermo-optic effect of ${\mathrm{Si}}_3{\mathrm{N}}_4$ for thermal control, achieving comb line spacing that is adjustable to integer multiples of the FSR (∼1.812 THz, 2.718 THz, and 3.624 THz) for micro-combs, which provides a solution for selecting spectral lines for a multi-wavelength optical source. Compared with the original FSR (∼0.906 THz) of the silicon nitride micro-comb, this dynamic adjustment of the spectral distribution offers capabilities for reconfigurability, which can improve spectral efficiency and frequency selectivity. Thus, the ability to dynamically modulate the output optical spectrum of the micro-comb can be leveraged for silicon photonic components like a reconfigurable integrated transmitter, where the power control of the output signal spectral components is essential. Therefore, our proposal, as an effective improvement solution for a wavelength-division multiplexing system, has the potential to address the requirements for high-repetition rates and a tunable multi-channel optical source in optical communications, photonic computing, optical networks, and optical interconnections in the future, offering significant advantages in enabling the parallel processing of large datasets and improving the bandwidth efficiency of optical networks. However, several challenges, such as optimizing power efficiency, managing thermal effects, and achieving high-precision fabrication, need to be addressed. By exploring efficient nonlinear materials to minimize power consumption and designing more robust thermal management techniques, these challenges will be overcome, thus enabling our proposal to play a pivotal role in shaping scalable, high-performance photonic systems and paving the way for advances in both photonic computing and next-generation optical communications.