Thermally Tunable Cascaded Silicon Microring Notch Filters: A Reproducible Transfer-Matrix and Robustness Study

Technical Summary: Thermally Tunable Cascaded Silicon Microring Notch Filters

Problem Statement
Silicon microring resonators are compact, wavelength-selective elements essential for integrated photonics, yet their performance is highly sensitive to temperature fluctuations, fabrication dimensional errors, and coupling variations. While single-ring effects are well-understood, cascaded rings introduce complex interactions where correlated and uncorrelated perturbations can reshape composite spectral notches in non-obvious ways. A primary challenge in the field is that many computational studies focus solely on optimizing a "nominal" spectrum. However, a design that achieves deep extinction at a single exact set of parameters may fail in practice if small deviations in coupling coefficients or effective indices destroy the notch. There is a need for a design methodology that quantifies robustness and sensitivity distributions rather than relying on a single optimized curve.

Methodology
This study employs a fully reproducible, compact transfer-matrix model to analyze a two-ring silicon notch filter in the 1550-nm band. The approach separates systematic thermal tuning from random fabrication-like perturbations.

• Physical Model: The model treats the cascaded system as the product of two independent single-ring transfer functions, neglecting back-reflection and coherent inter-ring feedback. It incorporates round-trip attenuation, bus-ring self-coupling ($t$), radius mismatch ($R_1$ vs. $R_2$), and a linearized effective index model that includes thermo-optic response and independent index perturbations ($\delta_n$).
• Parameter Space: The analysis evaluates a nominal design ($R_1=10.00$ $\mu$m, $R_2=10.08$ $\mu$m, $t=0.94$) across a temperature range of 290–330 K, coupling amplitudes of 0.90–0.98, and second-ring radii of 10.02–10.14 $\mu$m.
• Numerical Analysis:
  • Thermal Sweep: A linear least-squares fit determines the thermal wavelength sensitivity.
  • Design Mapping: A 2D map of extinction ratio is generated by sweeping coupling and radius to identify robust operating regions.
  • Monte Carlo Simulation: A 1,000-realization ensemble (seeded with random seed 11082) propagates Gaussian uncertainties in radius ($\sigma=8$ nm), coupling ($\sigma=0.006$), and effective index ($\sigma=2\times10^{-4}$) to quantify the statistical spread of notch wavelength, extinction ratio, linewidth, and loaded quality factor.
• Reproducibility: All numerical outputs, raw CSV data, and the Python code used for extraction are provided to ensure the study is auditable and independent of proprietary software.

Key Results

• Thermal Response: The composite notch translates linearly with temperature, exhibiting a fitted thermal sensitivity of 40.1 pm/K over the 290–330 K range. This shift is systematic and primarily translates the spectral structure rather than altering the relative alignment of the two rings.
• Coupling and Radius Dependence: Deep extinction does not occur at a single optimal point but is localized to specific combinations of coupling and radius that satisfy both spectral overlap and critical attenuation balance. The study reveals that the region of deep extinction is narrower than what nominal wavelength alignment alone would suggest.
• Robustness Analysis: The Monte Carlo study demonstrates that fabrication uncertainties significantly impact performance. Across 1,000 realizations, the extinction ratio showed a median of 0.41 dB with a 5th–95th percentile range of 0.02–0.51 dB, while the notch wavelength exhibited a standard deviation of 1.3471 nm. Crucially, wavelength error and extinction loss were found to be non-interchangeable; some realizations retained depth but shifted spectrally, while others remained spectrally aligned but lost depth.

Significance and Claims
The paper positions itself as a computational baseline study rather than an experimental validation or a foundry-qualified prediction. Its primary significance lies in:

1. Establishing a Reproducible Baseline: By providing complete code, fixed random seeds, and raw data, the work creates a defensible, transparent reference for future full-wave simulations or experimental comparisons.
2. Decoupling Design Dimensions: The study explicitly separates thermal control (a systematic shift) from fabrication tolerance (differential errors), arguing that a global tuning mechanism cannot necessarily correct for differential fabrication errors that degrade extinction.
3. Shifting Design Objectives: The author advocates for a shift in design philosophy from maximizing nominal extinction to optimizing for tolerance-aware objectives, such as maximizing the 5th-percentile extinction subject to wavelength constraints.
4. Methodological Efficiency: The work demonstrates that compact analytical models can effectively screen large parameter domains and expose dominant sensitivities, serving as a cost-effective precursor to expensive full-wave simulations or fabrication.

The paper concludes that while the specific numerical values are representative rather than process-calibrated, the identified tradeoffs between notch depth, linewidth, thermal sensitivity, and manufacturing uncertainty provide a robust framework for designing manufacturable silicon photonic filters.