Silicon photonics is one of the most promising technologies to enable low-cost, low power consumption and high-performance optical transceivers. Compared to Mach-Zehnder modulators (MZMs), silicon microring modulators (Si-MRMs) have attracted significant attention in recent years due to their compact footprint, high modulation speed, and potentially more energy-efficient dense integration for multi-lane data transceivers. However, Si-MRMs are highly sensitive to fabrication process variations and environmental fluctuations, leading to resonance drift and changes in coupling states that degrade signal performance, especially for advanced modulation formats like 4/8-level pulse amplitude modulation (PAM4/PAM8). Despite this, there is a lack of in-depth quantitative analysis on the effect of modulator coupling states on system performance. Typically, MRM parameters such as radius, coupling gap, doped regions, metal contacts, and waveguide dimensions are carefully calculated and chosen to achieve critical coupling. However, during fabrication, testing, or deployment in communication systems, the coupling state of the Si-MRM can shift from critical coupling to overcoupling or undercoupling due to fabrication errors, temperature fluctuations, and applied voltage, affecting system performance. A quantitative investigation into the influence of Si-MRM coupling states on system performance is crucial for modulator design and integration into various systems.

In this study, we conduct a comprehensive quantitative analysis of the effect of Si-MRM coupling states on a high-speed PAM4 transmission system using a system-level model that includes a dynamic ring resonator model and an equivalent electrical circuit. Performance metrics such as signal bit error rate (BER), receiver side maximum received optical power (RoP), power penalty, device bandwidth, and eye diagram are investigated. Modeling and simulations are carried out in VPI Transmission Maker and Matlab. The dynamic ring resonator model simulates the optical properties of modulators, capturing variations in the optical field within the resonant cavity coupling region and at the input and output ports over voltage and time, as well as adjustments in signal phase and power within the ring waveguide. The circuit subsystem models the effect of voltage on parameters such as resistor, junction capacitance, and inductance, including equivalent circuitry for wire bonding. The MRM model is made of silicon with a depletion-type phase shifter. In the simulation, the MRM radius is set to 10 μm, corresponding to a free spectral range (FSR) of 8.75 nm. The lateral PN junction in the ring waveguide provides varying carrier depletion at different reverse voltages. Doping concentrations for n and pin the low-doped region are 3.5×10¹⁸ cm^-3 and 6.5×10¹⁸ cm^-3, respectively, while the high-doped region features a doping concentration of 4.5×10²¹ cm^-3 for both n and p. The measured loaded Q factor of the MRM is ~4655 at a bias voltage of 0 V. The electro-optic (EO) phase efficiency of the PN junction is measured to be 0.56 V·cm at a reverse bias voltage of 2 V. In the simulation, the coupling states of the microring resonator are regulated by manipulating the coefficients a and t, which correspond to the single-pass amplitude transmission factor and self-coupling coefficient, respectively.

The simulation results indicate that for PAM4 generation operating under critical coupling when the data rate of the generated signal is below 150 Gbit/s, incremental variations of a and t from 0.59 to 0.91 lead to a maximum power penalty of 4.2 dB. Optimal system performance is attained when values of a and t range from 0.71 to 0.83. When the modulation speed exceeds 160 Gbit/s and a(t) value is set to 0.91, significant enhancement in system performance can be attained due to increased device bandwidth. In overcoupling and undercoupling states, where the modulation speed remains below 150 Gbit/s and adequate device bandwidth is maintained, variations in t with a constant a or adjustments in a with a constant t result in degraded system performance compared to the critical coupling state. Only slight variations in t (ranging from 0.71 to 0.79, with a maintained at 0.75) or a (ranging from 0.71 to 0.79, with t maintained at 0.75) can yield system performance closely approximating that of the critical coupling state (where a and t are both set at 0.75). When the modulation speed surpasses 180 Gbit/s, achieving BER performance below the threshold of soft-decision forward error correction (SD-FEC) is possible only by increasing the value of a to 0.83?0.91 to enhance the bandwidth. Under these conditions, system performance exceeds that of critical coupling. However, in the undercoupling state, no results have been observed where performance exceeds that achieved under critical coupling conditions.

Our study provides a comprehensive and quantitative analysis of the effect of coupling states of Si-MRM on the generation of high-speed PAM4 signals (ranging from 64 Gbit/s to 224 Gbit/s). The entire analysis is based on a system-level model of Si-MRM. In the simulation, different coupling states are simulated by changing the values of a and t, and a quantitative analysis is conducted on key performance indicators, including BER performance, RoP at the receiver, power penalties, device bandwidth, and eye diagrams. The results show that optimal system performance is attained at the critical coupling state, and the bandwidth meets the rate requirement. Within this context, the best system performance is observed when a(t) ranges between 0.71 and 0.83. In scenarios of overcoupling or undercoupling states, where the modulation bandwidth aligns with the modulation rate requirement, variations in t with a constant a or adjustments in a with a constant t result in degraded system performance compared to the critical coupling state. Otherwise, performance below the threshold for BER with SD-FEC is possible only by increasing the value of a to enhance the bandwidth. In this scenario, the overcoupling state exhibits enhanced performance compared to the critical coupling state. However, no instances within the undercoupling state, which involve changes in a, have displayed performance better than that of the critical coupling state. Our study provides valuable insights into the quantitative assessment of Si-MRM coupling state variations caused by fabrication, testing, network deployment, etc., and their consequent effects on the performance alterations of high-speed modulation signals. Such findings are pivotal for directing the development of next-generation 800 Gbit/s和1.6 Tbit/s chip-level high-speed optical interconnects utilizing Si-MRM technology.