🔬 applied physics

Inverse-Designed Photonic Crystal Cavities with Controllable Far-Field Numerical Aperture

This paper presents an inverse design framework that simultaneously optimizes photonic crystal cavities for high quality factors and controllable far-field numerical apertures, experimentally demonstrating significant improvements in both coupling efficiency and quality factor compared to standard designs.

Original authors: Neelesh Kumar Vij, Purbita Purkayastha, Jasvith Raj Basani, Edo Waks

Published 2026-10-05

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CC BY 4.0

Original authors: Neelesh Kumar Vij, Purbita Purkayastha, Jasvith Raj Basani, Edo Waks

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

Light behaves in strange ways when it is squeezed into spaces smaller than its own wavelength. In the microscopic world of photonics, scientists build tiny structures called photonic crystal cavities to trap light in these tight spaces. Think of these cavities as high-tech mirrors arranged in a grid, creating a prison for photons that forces them to bounce around for a long time before escaping. This confinement is powerful because it makes light interact strongly with matter, a feature essential for building ultra-efficient lasers, sensitive sensors, and the hardware needed for future quantum computers. To make these devices useful, engineers need them to hold onto light for a long time without leaking, while also allowing that light to escape easily into a fiber optic cable when the time is right.

For years, improving these tiny light traps has been a difficult balancing act. Making a cavity hold light better often meant making it harder to get the light out, and vice versa. Furthermore, controlling exactly how the light beams out into the world—specifically the angle and spread of the beam—has been a major hurdle. Most previous designs could either trap light well or direct it well, but rarely both at the same time with precise control over the beam's shape. This limitation has slowed progress in fields that rely on capturing and manipulating single particles of light.

A team of researchers at the University of Maryland has now solved this problem using a method called inverse design. Instead of starting with a shape and trying to guess how it will perform, they started with the exact performance they wanted and let a computer work backward to find the shape that would create it. They set two specific goals for their computer: first, the cavity needed to trap light with extreme efficiency, and second, the light escaping from the cavity needed to form a beam with a specific, controllable spread. By adjusting the positions of tiny holes in a thin sheet of silicon nitride, the computer generated four unique cavity designs. Each design was tailored to produce a light beam with a different spread, known as a numerical aperture, ranging from a tight, focused beam to a wider, more open one.

The researchers then built these computer-generated designs in a laboratory. They took a thin layer of silicon nitride and used an electron beam to etch patterns of holes into it, creating the physical cavities. When they shone a laser on these structures, the results matched their predictions. The new cavities trapped light significantly better than standard designs, with the quality of the light storage improving by more than three times. More importantly, the light that escaped did so with a much higher efficiency. The best-performing new design captured and directed light nearly twenty-eight times better than the traditional standard design. This massive improvement means that scientists can now collect light from these tiny sources much more easily, which is a critical step for making quantum devices practical.

The team also tested how well these designs would hold up against the tiny imperfections that inevitably happen during manufacturing. Real-world factories cannot make holes with perfect precision; the edges are never perfectly smooth, and the sizes vary slightly. The researchers analyzed the actual holes in their fabricated devices and found that while these small errors did reduce the performance, the new designs remained robust. Even with these manufacturing flaws, the cavities still outperformed older designs by a wide margin. This suggests that the method is not just a theoretical success but a practical one that can survive the realities of mass production.

By proving that they can simultaneously optimize how well a cavity holds light and how it releases that light, the researchers have opened a new path for photonic engineering. Their work demonstrates that it is possible to design light traps that are both high-performing and easy to connect to the outside world. This advancement could lead to more efficient optical chips, better sensors, and more reliable components for quantum information systems, all built on a foundation of light that is finally under precise control.

Technical Summary: Inverse-Designed Photonic Crystal Cavities with Controllable Far-Field Numerical Aperture

Problem Statement
Photonic crystal cavities are essential for low-power photonics, nonlinear optics, and cavity quantum electrodynamics due to their ability to confine light in subwavelength volumes, enhancing light-matter interactions. However, realizing practical applications requires a simultaneous optimization of three critical metrics: high quality factors ( $Q$ ), low mode volumes, and high coupling efficiencies to external optics. Traditional design approaches face significant challenges:

Design Space Complexity: The exponential growth of the design space with increasing parameters makes manual optimization impractical.
Trade-offs in Existing Methods: While inverse design has been used to optimize $Q$ and mode volume, previous efforts often sacrificed coupling efficiency or lacked the ability to precisely control the far-field radiation pattern.
Far-Field Control: Applications such as coupling to fibers or nanophotonic waveguides demand a specific Gaussian profile with a controllable numerical aperture (NA). Existing heuristic methods (e.g., band folding) often improve directionality at the expense of $Q$ , while other directional designs struggle to achieve precise NA matching.

Methodology
The authors present an inverse design framework based on gradient-based optimization to simultaneously maximize the cavity quality factor and tailor the far-field numerical aperture.

Simulation Engine: To navigate the large design space efficiently, the authors utilize the Guided Mode Expansion (GME) method rather than computationally intensive Finite-Difference Time-Domain (FDTD) simulations for the optimization loop. GME approximates cavity modes using eigenmodes of a homogeneous permittivity slab. The open-source Python package Legume is used, which supports automatic differentiation for efficient gradient computation.
Optimization Variables: The optimization adjusts the positional displacements of holes within the first quadrant of an L3 photonic crystal cavity. Symmetry is enforced by applying these displacements to the other quadrants. Crucially, hole radii are excluded from optimization to ensure the final designs remain compatible with standard nanofabrication capabilities (avoiding the difficulty of fabricating varying hole sizes).
Cost Function: A multi-objective loss function $L(Q, \eta)$ $L (Q, η)$ is minimized, where $Q$ $Q$ is the quality factor and $\eta$ $η$ is the coupling efficiency. The coupling efficiency is defined as the overlap integral between the cavity's transverse far-field mode and a target outgoing Gaussian beam characterized by a specific numerical aperture ( $NA_t$ $N A_{t}$ ).
- The loss function includes a term to normalize the contribution of $Q$ relative to a target $Q_t$ .
- A spatial penalty mask $W$ in reciprocal space penalizes far-field emission exceeding the target NA, forcing the optimizer to suppress high-angle radiation and converge toward a single Gaussian-like lobe.
Target Designs: Four distinct L3 cavities (labeled NA2, NA3, NA4, NA5) were designed with target NAs of 0.2, 0.3, 0.4, and 0.5, respectively, all targeting a quality factor of $10^5$ in Silicon Nitride (SiN) at a resonance wavelength of $\approx 520$ nm.

Key Results
The study validates the framework through both simulation and experimental fabrication in SiN.

Simulation Performance:
- The inverse-designed cavities achieved simulated quality factors ranging from $4.4 \times 10^4$ to $6.1 \times 10^4$ , representing more than an order of magnitude improvement over standard L3 cavities ( $1.3 \times 10^3$ ) and band-folding designs ( $5.0 \times 10^2$ ).
- Coupling efficiencies ( $\eta_0$ ) were calculated to be $>0.6$ for NA2, NA3, and NA4, and 0.37 for NA5.
- The Purcell factors for the inverse-designed cavities reached up to 4772, a 49-fold increase over the standard L3 cavity.
- Comparisons with a directional cavity (D1) and band-folding (BF) designs showed that the inverse-designed cavities offered superior simultaneous performance in $Q$ and coupling efficiency.
Experimental Validation:
- Cavity designs were fabricated using electron-beam lithography and reactive dry etching on SiN membranes.
- Photoluminescence measurements at room temperature confirmed the ability to control the far-field NA. The NA4 design ( $NAt=0.4$ ) demonstrated the best mode-matching with the collection optics (NA=0.29), yielding the highest coupling efficiency.
- Quantitative Improvements: Compared to the initial standard L3 design, the optimized NA4 cavity exhibited a 3.4-fold increase in quality factor and a 27.4-fold improvement in coupling efficiency.
- Statistical analysis across 8–9 devices per design confirmed that these performance gains are robust and reproducible, not limited to isolated devices.
Disorder Analysis:
- Scanning Electron Microscopy (SEM) analysis revealed a hole radius disorder standard deviation of $\sigma_r \approx 1.8$ nm.
- Simulations incorporating this disorder showed that while performance degrades (median $Q$ and $\eta$ dropping to 30–60% of design values), the inverse-designed cavities retain significant performance.
- The discrepancy between simulated and measured $Q$ values was attributed primarily to non-vertical sidewall angles rather than hole radius disorder.

Significance and Claims
The authors claim this work represents a significant step toward the multi-objective optimization of photonic crystal cavities. Key contributions include:

Simultaneous Optimization: Demonstrating a framework that successfully optimizes for high $Q$ and a user-specified far-field NA simultaneously, overcoming the trade-offs inherent in previous heuristic or single-objective designs.
Fabrication Compatibility: The method relies solely on hole displacement, avoiding complex variable-radius features, thereby ensuring the designs are manufacturable with standard nanofabrication techniques.
Experimental Control: Providing the first experimental evidence that the far-field NA of photonic crystal cavities can be controllably tailored via inverse design while maintaining high quality factors.
Practical Impact: The substantial improvement in coupling efficiency (27.4-fold) implies that quantum emitters can be accessed with significantly lower excitation powers, potentially reducing laser-induced dephasing and improving spin coherence. Additionally, efficient photon extraction accelerates N-photon correlation measurements by reducing acquisition times.

The paper concludes that this versatile framework can be extended to other photonic crystal architectures (e.g., H1, L4/3, heterostructures) and holds promise for applications in cavity quantum electrodynamics, optoelectronics, and display technologies.

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