Resolving Complex Subwavelength Grating Structures Using Topologically Structured Light

Technical Summary: Resolving Complex Subwavelength Grating Structures Using Topologically Structured Light

Problem Statement
While far-field optical imaging has advanced beyond the classical Abbe diffraction limit (~λ/2) through various signal processing and structured light techniques, significant challenges remain in retrieving dimensional parameters from arbitrarily structured objects. Previous successes in optical metrology have largely been confined to tightly constrained problems, such as determining the position of a single nanowire or the widths of a double slit. The authors investigate whether the advantages of topologically structured illumination—specifically its ability to enhance measurement precision via phase singularities—can be extended to the more complex task of retrieving the dimensional profiles of arbitrary, random binary gratings. As the dimensionality of the parameter space increases and feature sizes decrease below the diffraction limit, the inverse scattering problem becomes significantly more difficult.

Methodology
The study employs a computational approach combining superoscillatory illumination with deep learning to solve the inverse scattering problem for random one-dimensional binary gratings.

• Object Generation: A dataset of 4,096 unique grating profiles was generated. Each profile is 10 μm long with 10 nm pixelation, consisting of random sequences of transparent and opaque states. This design ensures feature sizes span a range from approximately λ/2 down to λ/50 (where λ = 488 nm), with uncorrelated transitions to simulate arbitrary complexity.
• Illumination Schemes: Two illumination conditions were compared:
  1. Plane Wave: Unstructured, uniform illumination.
  2. Topologically Structured Light: An axially symmetric superoscillatory field formed by the linear combination of two band-limited prolate spheroidal wave functions (PSWFs). This field features a central intensity peak (0.3λ half-maximum width) flanked by phase singularities, creating strong local subwavelength intensity and phase variations.
• Simulation: Transmission scattering patterns were calculated at a distance of 2λ from the sample using the angular spectrum method. The goal was to retrieve the dimensional profile of the central 1 μm section of the grating (represented as a 1×100 vector of 0s and 1s).
• Neural Network Architecture: A convolutional neural network (CNN) was trained to map scattering patterns to grating profiles. The architecture included two convolutional layers (4×4 and 2×2 kernels) followed by pooling layers (3×3 and 2×2 kernels), and two fully connected layers (128 and 100 outputs).
• Training and Testing: The dataset was split into training/validation (2,048 patterns) and testing (2,048 unseen patterns) sets. Results were aggregated over ten independent training cycles.
• Multi-Shot Extension: To test positional sensitivity, a "few-shot" approach was simulated where diffraction patterns were recorded for three beam positions: centered (0,0) and displaced by ±Δx (~λ/3).

Key Contributions and Results
The paper demonstrates that topologically structured illumination, when coupled with deep learning, enables the retrieval of sub-diffraction features in arbitrary objects with higher precision than plane wave illumination.

• Single-Shot Resolution: Under superoscillatory illumination, the neural network successfully retrieved feature sizes down to ~λ/7 (70 nm) with a >95% success rate. This represents a 1.4× improvement over plane wave illumination, which resolved features down to ~λ/4.9 (100 nm).
• Multi-Shot Resolution: By utilizing three diffraction patterns (centered and ±Δx displaced), the resolution for superoscillatory illumination improved to ~λ/10.5. Crucially, the plane wave case showed no improvement with positional displacement, as its scattering patterns are invariant to translation. This yields a 2.2× advantage for the structured light approach in few-shot imaging.
• Complexity and Interference: Counterintuitively, retrieval success rates were found to be higher for more complex objects (those with more features) compared to simple structures (1–3 features). The authors attribute this to interference effects; as the number of scattering features increases, the total information content of the scattered field increases, aiding the neural network's deconvolution process. This effect is more pronounced in plane wave illumination but persists in the structured light regime.
• Learning Dynamics: The increased complexity of the scattering patterns under superoscillatory illumination required more training epochs for the network to converge. However, this resulted in a lower mean squared error and higher retrieval success for smaller feature sizes compared to the plane wave case.

Significance and Claims
The paper claims to demonstrate that the advantages of topologically structured illumination, previously established for constrained metrology tasks (e.g., single nanoparticle tracking), are retained and effective when the parameter space expands to include the retrieval of unspecified dimensional parameters from complex, arbitrary objects.

The authors assert that the interaction between the strong local intensity and phase variations of the superoscillatory field and the nanometric features of the object generates scattering patterns with significantly higher information content. Deep learning algorithms can leverage this enhanced information to perform a deconvolution that resolves features well below the classical diffraction limit. The work establishes a path toward using topologically structured light for increasingly complex nanoscale optical metrology and basic binary object imaging, achieving single-shot resolution of ~λ/7 and multi-shot resolution of ~λ/10.5.