Inverse designed photonic crystal waveguides for pulsed operation: dispersion, losses, and controlled light-matter interactions

Technical Summary: Inverse Designed Photonic Crystal Waveguides for Pulsed Operation

Problem Statement
Photonic crystal waveguides (PCWs) offer sub-wavelength light confinement and the ability to manipulate group velocity, making them ideal for enhancing light-matter interactions. However, their application in pulsed optical systems is hindered by two primary challenges: dispersion and disorder-induced losses. In typical PCWs, different frequency components of a pulse propagate at different velocities due to dispersion, causing pulse broadening and reshaping. Furthermore, as the group index ($n_g$) increases to achieve slow light, backscattering losses due to fabrication disorder (roughness at hole edges) scale with $n_g^2$, severely limiting device length and bandwidth. Previous dispersion engineering techniques, often relying on brute-force parameter sweeps or limited optimization methods, struggle to simultaneously achieve large bandwidths, constant group velocity, and low loss.

Methodology
The authors propose a gradient-based inverse design framework that significantly accelerates the optimization process. The core of their methodology involves:

1. Efficient Mode Solving: Utilizing the Guided Mode Expansion (GME) method to solve for Bloch modes and their associated frequencies. This approach is reported to be over 100 times faster than standard full 3D Finite-Difference Time-Domain (FDTD) or finite-element solvers.
2. Physics-Based Formulas: Incorporating analytical formulas for backscattering loss derived from perturbation theory and Green's function scattering theory. This allows for the efficient calculation of disorder-induced loss without full 3D scattering simulations.
3. Constrained Optimization: The design problem is formulated as a constrained minimization of a cost function. The objective is to maintain a target group index ($n_g^t$) across a specified bandwidth (defined by the Normalized Delay-Bandwidth Product, NDBP).
4. Constraints: The optimization enforces several practical constraints:
   • Loss: A maximum acceptable backscattering loss, normalized to account for $n_g$ dependence.
   • Frequency Range: Ensuring the mode remains within a specific frequency window.
   • Single-Mode Operation: Preventing overlap with neighboring bands to avoid inter-mode scattering.
   • Fabrication: Enforcing minimum hole radii and inter-hole distances to ensure manufacturability.
5. Algorithm: The constrained problem is solved using a trust-region constrained optimization algorithm (SciPy's trust-constr), which handles the trade-off between the cost function and constraints via a barrier method.

Key Contributions and Results
The study performs a systematic exploration of the trade-offs between bandwidth, group index, dispersion, and loss across a vast design space (25,500 separate optimizations).

• Performance Gains: The inverse design approach yields PCWs with up to an order-of-magnitude increase in bandwidth and up to a 4-fold reduction in loss compared to conventional designs. For example, for a target $n_g=30$, the bandwidth increases from 0.5 nm to 25 nm, while loss decreases from 36 dB/cm to 9 dB/cm.
• Trade-off Analysis: The authors map the inherent trade-offs between dispersion and normalized loss. They demonstrate that while increasing the NDBP or group index generally shifts the performance curve toward higher dispersion and loss, optimized designs can still significantly outperform standard W1 waveguides.
• Application 1: Quantum Photonics: The method is applied to design PCWs for quantum dot (QD) interfaces. The optimization targets a high Purcell factor ($PF$) within a specific spatial region (center of the waveguide) while maintaining low loss and broad bandwidth. Results show designs capable of supporting a 5 nm bandwidth with a Purcell factor $>5$, a significant improvement over the ~0.2 nm bandwidth of standard designs for equivalent performance. This addresses the challenge of inhomogeneous broadening in quantum emitters.
• Application 2: Optical Communications: The framework is used to design compact phase shifters for Mach-Zehnder Modulators (MZMs). By optimizing for a specific phase shift per unit length while managing doping-induced losses, the authors demonstrate designs with bandwidths up to 15 nm (compared to 1 nm for standard designs). The optimized devices achieve a $\pi$-phase shift length ($L_\pi$) as short as 0.26 mm with losses as low as 1.9 dB/cm, offering a trade-off space between compactness and loss.

Significance
The paper claims that this work provides a general, efficient method for optimizing PCW integrated technologies. By combining GME with gradient-based inverse design, the authors overcome the computational bottlenecks that previously limited the exploration of complex design spaces. The resulting designs offer substantial improvements in bandwidth and loss, enabling more robust pulsed operation and facilitating practical applications in quantum light-matter interactions and high-speed optical communications. The authors emphasize that their approach reveals the fundamental limits and trade-offs of PCW engineering, providing a pathway to application-specific devices with best-in-class specifications. The methodology relies on open-source software, suggesting it is adaptable to various seed geometries and design problems.