🔬 optics

Multimode grating couplers via foundry-compliant inverse design

This paper presents a systematic inverse design framework for creating robust, foundry-compliant multimode grating couplers that maintain high efficiency despite manufacturing imperfections, with minimum feature size identified as the critical constraint for optimization.

Original authors: Hao Li, Nazar Pyvovar, Zhaowei Dai, Owen D. Miller

Published 2026-05-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hao Li, Nazar Pyvovar, Zhaowei Dai, Owen D. Miller

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a complex, multi-lane highway system on a tiny piece of silicon. Your goal is to take beams of light (like cars) coming from the open sky and guide them smoothly onto specific lanes (waveguides) on a computer chip. This is what grating couplers do.

However, building these highways is tricky. You can't just draw any shape you want; you have to follow strict rules set by the factory (the "foundry") that makes the chips. The biggest rule is the Minimum Feature Size (MFS): you cannot build any part of the road narrower than a certain width (like 62 nanometers). If you try to build a turn that is too sharp or a lane that is too thin, the factory machinery can't make it, and the design fails.

This paper presents a new, smart way to design these light highways that respects the factory's rules while still letting a lot of traffic through. Here is how they did it, explained simply:

1. The Problem: The "Too Small" Rule

In the past, scientists used computers to design these light couplers. The computers would come up with incredibly complex, jagged shapes that were perfect for moving light but had tiny, microscopic details that factories couldn't actually build. It was like designing a sculpture with a pencil tip that is too fine for the chisel the factory has.

The authors focused on visible and near-infrared light (the colors we can see and the ones used in fiber optics). At these wavelengths, the "minimum feature size" rule is very strict because the light waves are so small. If your design requires a feature smaller than the factory allows, the whole thing breaks.

2. The Solution: A Four-Step "Sculpting" Process

The team developed a systematic, four-step "inverse design" workflow. Think of this as sculpting a statue, but instead of clay, they are sculpting with light and math.

Step 1: The "Ghost" Sketch (Grayscale Optimization):
First, they let the computer design the highway without worrying about the factory rules. They allow the material to be "fuzzy" or "gray" (partly air, partly solid). This lets the computer find the absolute best, most efficient shape, even if it looks like a messy, impossible blob.
Step 2: Making it Real (Binarization):
Factories can only make two things: solid silicon or empty air. There is no "gray." The team slowly forced the computer to turn that fuzzy "gray" design into a sharp, black-and-white (solid or air) design. It's like taking a watercolor painting and turning it into a high-contrast stencil.
Step 3: Enforcing the Rules (Minimum Feature Size):
Now, they applied the factory's "no tiny parts" rule. The computer looked at the design and said, "This turn is too narrow!" and automatically merged or removed the tiny bits that were too small. This step usually hurts performance (like having to take a wider, less direct route), so the efficiency drops.
Step 4: Fine-Tuning the Edges (Shape Optimization):
Finally, they treated the boundaries between the solid and air like movable walls. They nudged these walls slightly to recover the lost efficiency, ensuring the design was still perfect for moving light but now obeyed the factory's size limits.

3. The Surprising Discovery: "Fill the Holes"

One of the most interesting findings was a counter-intuitive trick. Usually, to move light efficiently, you want a big difference between the materials (like solid silicon vs. empty air). This is like having a steep hill; it's easy to roll a ball down.

However, when the factory forces you to make the "hills" wider (larger minimum feature size), the steep hills become too aggressive and chaotic. The authors found that filling the empty air holes with a material like glass (SiO₂) actually helped.

The Analogy: Imagine trying to guide a crowd of people through a maze. If the walls are too sharp and the gaps are too narrow, people get stuck. If you soften the walls (by filling the air with glass), the crowd moves more smoothly, even if the gaps are wider.
The Result: By reducing the difference between the materials, they could actually achieve better efficiency with larger, factory-friendly features than they could with the "perfect" but unbuildable designs.

4. Making it Robust: The "Wobbly Table" Test

Even if a design fits the factory rules, it might fail if the factory isn't perfect. Machines can over-etch (cut too deep) or under-etch (cut too shallow).

The team tested their designs by simulating these errors. They found that their designs were very sensitive to "over-etching" (like a table that wobbles if one leg is slightly too short).
They added a "robustness" step to the design process. They told the computer: "Design it so that even if we cut it 10% too deep or 10% too shallow, it still works."
The Result: The final designs were slightly less efficient in a perfect world, but they were incredibly stable in the real world. They could handle the factory's mistakes without the light getting lost.

Summary of Results

What they built: Designs that can take four different beams of light (at different colors or angles) and guide them onto a chip simultaneously.
How well it works: They achieved 50–60% efficiency (meaning more than half the light gets to the right place), even with strict factory rules.
Why it matters: This proves you can build complex, multi-function light highways on a chip using standard factory equipment, without needing expensive, custom-made tools.

In short, the paper shows how to use smart math to design light-guiding structures that are not only efficient but also "factory-ready" and tough enough to survive the messy reality of manufacturing.

1. Problem Statement

Grating couplers are essential interfaces for coupling free-space optical beams into on-chip waveguide modes, critical for applications in integrated photonics, optical interconnects, and quantum computing. While single-function couplers are well-understood, designing multifunctional couplers (simultaneously coupling multiple wavelengths, angles, or polarizations) that satisfy strict foundry constraints remains a significant challenge.

Key challenges addressed in this work include:

Multimode Complexity: Simultaneously coupling multiple independent waves (e.g., 4 wavelengths or 2 wavelengths $\times$ 2 angles) requires complex phase matching and apodization.
Foundry Constraints: Modern semiconductor fabrication imposes strict Minimum Feature Size (MFS) limits (typically 50–100 nm for visible/near-infrared). These limits are particularly restrictive at shorter wavelengths (visible to near-IR) where the feature size represents a large fraction of the wavelength.
Fabrication Imperfections: Real-world manufacturing introduces errors such as over/under-etching, critical dimension (CD) variations, overlay misalignment, and sidewall angle deviations. Many high-efficiency inverse-designed structures are fragile and lose performance under these variations.
Lack of Scaling Laws: There is limited systematic understanding of how MFS constraints and fabrication tolerances affect the performance scaling of multimode couplers.

2. Methodology

The authors propose a systematic, four-stage inverse design workflow based on gradient-based topology optimization to discover foundry-compliant, high-efficiency multimode grating couplers.

The Four-Stage Workflow:

Grayscale Topology Optimization:
- The design begins with a continuous "grayscale" density variable $\rho(x)$ representing material distribution (e.g., SiN vs. air).
- A gradient-based optimizer (Method of Moving Asymptotes, MMA) maximizes the Figure of Merit (FOM), defined as the average coupling efficiency across all target modes.
- Adjoint methods are used to compute gradients efficiently.
Progressive Binarization:
- To ensure the design is fabricable (binary materials only), the grayscale design is projected onto a binary state using a smoothed Heaviside function (modified hyperbolic tangent).
- A "subpixel-smoothed projection" (SSP) method is employed to maintain gradient accuracy as the design becomes binary.
- The steepness parameter $\beta$ is ramped up gradually through a specific schedule to avoid trapping the optimizer in poor local minima.
Enforcement of Minimum Feature Size (MFS):
- Geometric constraints are applied to ensure no feature (solid or void) is smaller than the target MFS.
- This is achieved using eroded and dilated realizations of the design via density filtering. Two inequality constraints ( $g_s$ and $g_v$ ) ensure that the distance between solid and void phases never drops below the MFS.
- These constraints are applied after the design has largely converged to a binary state to avoid trapping the optimizer early in the process.
Edge-Based Shape Optimization:
- Once the topology is fixed and binary, the design variables are converted to explicit edge positions (boundaries between materials).
- A final shape optimization fine-tunes these boundaries at sub-pixel resolution to recover performance lost during the MFS enforcement, while strictly maintaining the MFS and binary constraints.

Robustness Optimization:

To address fabrication imperfections, the authors introduce a robust optimization step.
Instead of optimizing only for the nominal geometry, the FOM is calculated as a weighted average of the nominal, over-etched, and under-etched configurations.
This forces the design to settle in a "flat" region of the performance landscape, making it insensitive to systematic etch errors.

Simulation Setup:

Material: Silicon Nitride (SiN) core with Air or SiO $_2$ cladding.
Wavelengths: Visible (~~600 nm) and Near-Infrared (~~900 nm).
Solver: Volume Integral Equation (VIE) method accelerated by a fast direct solver.
Target: Coupling 4 independent waves (e.g., 4 wavelengths or 2 wavelengths $\times$ 2 angles) into waveguide modes.

3. Key Contributions

Foundry-Compliant Multimode Design: Demonstrated the first systematic framework to design multimode grating couplers that strictly adhere to MFS constraints (down to 62 nm) while maintaining high efficiency.
Robustness to Etch Errors: Showed that optimizing for systematic over/under-etch errors yields designs that are inherently robust to other imperfections (random CD variations, overlay, sidewall angles) with minimal efficiency penalty (<5%).
Counter-Intuitive Index Contrast Finding: Discovered that reducing the refractive index contrast (filling air holes with SiO $_2$ $_{2}$ ) can actually improve coupling efficiency when MFS constraints are tight.
- Mechanism: High index contrast (SiN/Air) leads to strong scattering that is difficult to control with large feature sizes. Lower contrast (SiN/SiO $_2$ ) weakens scattering per unit, allowing for better wavefront control and reduced sensitivity to dimensional errors.
Scaling Laws: Established that the primary determinant of performance trade-offs is the total number of input-output mode pairs, regardless of whether they are distinguished spectrally or spatially.

4. Key Results

Performance:
- Achieved average coupling efficiencies of 50–60% for 4-function couplers (e.g., 4 wavelengths) at 900 nm with a 62 nm MFS.
- Specific example: 4 wavelengths (791, 827, 864, 900 nm) coupled to the fundamental mode with efficiencies of 45%, 70%, 73%, and 50% respectively.
MFS Trade-off:
- Efficiency decreases monotonically as MFS increases, but the decay is relatively slow.
- At 900 nm, a 62 nm MFS yields ~60% efficiency; at 600 nm, the same MFS yields lower efficiency due to the larger MFS/ $\lambda$ ratio.
Robustness:
- Nominal Design: Highly sensitive to etch errors; efficiency drops from ~60% to <40% with $\pm$ 10% CD etch error.
- Robust Design: Maintains >50% efficiency across the full $\pm$ 10% etch range. The peak nominal efficiency drops only slightly (~5%) to gain this robustness.
- The robust design also showed reduced sensitivity to random CD variations and overlay misalignment.
Material Platform Comparison:
- For MFS > 31 nm, the SiN/SiO $_2$ platform outperformed the SiN/Air platform.
- At MFS = 47 nm, SiN/SiO $_2$ achieved 52% mean efficiency vs. 35% for SiN/Air (a 1.5x improvement), validating the strategy of lowering index contrast to mitigate MFS penalties.

5. Significance

This work provides a practical, generalizable framework for high-efficiency multimode coupling within the constraints of modern semiconductor foundries.

Industrial Relevance: By proving that inverse-designed devices can survive realistic fabrication variations (etch, overlay, CD) and strict MFS rules, it bridges the gap between theoretical inverse design and industrial manufacturing.
Design Strategy: The finding that lower index contrast can be beneficial under tight MFS constraints offers a new design heuristic for visible and near-IR photonics, challenging the conventional wisdom that higher contrast is always better.
Scalability: The demonstrated scaling laws suggest that high-efficiency coupling is achievable even for complex, many-mode systems (hundreds to thousands of modes), paving the way for advanced quantum computing and optical interconnect architectures.

In summary, the paper establishes that through a rigorous, multi-stage inverse design workflow incorporating robustness and geometric constraints, it is possible to create high-performance, multimode grating couplers that are ready for fabrication in standard semiconductor foundries.