MetaEOL: Hybrid Photonics and ML Paradigms

• The paper demonstrates a differentiable metalens-enhanced optical design that co-optimizes subwavelength metasurface features with conventional lens parameters to reduce aberrations.
• The paper introduces a meta-task prompting method that extracts one-word LLM embeddings from multiple semantic tasks, achieving state-of-the-art performance on STS benchmarks.
• The paper details a static meta-optic device with a cubic phase profile that extends the depth of focus by 250×, enabling high-resolution varifocal imaging.

MetaEOL encompasses three distinct, technically advanced frameworks across photonics and machine learning, all united by the “MetaEOL” acronym but serving separate domains: (1) Metalens-Enhanced Optical Lens design for hybrid wave-ray optical systems (Zhu et al., 2023), (2) Meta-Task Prompting for eliciting high-quality LLM embeddings (Lei et al., 2024), and (3) meta-optic extended depth of focus devices for varifocal imaging (Whitehead et al., 2021). This article systematically addresses these three “MetaEOL” paradigms in detail, reflecting their original contributions to imaging systems and neural representation learning.

1. Metalens-Enhanced Optical Lens (MetaEOL) for Differentiable Hybrid Optics

MetaEOL denotes a fully differentiable computational framework integrating a thin, flat metalens (metasurface with arbitrary phase/amplitude control) in front of a conventional refractive lens. This hybrid architecture allows simultaneous, gradient-based co-optimization of metalens subwavelength features and macroscopic lens parameters to engineer optical systems that combine the wide-ranging, phase-engineering capacity of metasurfaces with the computational scalability and long focal length achievable with standard ray optics (Zhu et al., 2023).

Traditional methods are limited by a dichotomy: wave-optical solvers scale poorly to macroscopic apertures but capture crucial physical effects (diffraction, field propagation), while classical ray-tracing is efficient but fails to model key aberrations and wave phenomena. The MetaEOL design addresses these issues by employing a differentiable graph that links local Maxwellian phase design (via surrogate RCWA-SIREN network mappings for meta-atom geometry) and vectorized GPU ray-tracing for the refractive element. The modeling pipeline is outlined as follows:

1. Input Synthesis: Generate input wavefronts $\varphi_{in}$ using plane waves or point source models.
2. Metalens Modulation: Modulate the field via $\varphi_{mod}(r) = A(r)\exp[j S(r)]$, with $A(r)$ and $S(r)$ supplied by a differentiable neural surrogate trained on meta-atom simulations. Output field: $$\varphi_{out}(r) = \varphi_{in}(r) \cdot \varphi_{mod}(r)$$.
3. Wave-to-Ray Conversion: Differentiably extract phase-gradient rays $k(r) = \nabla_r S(r)$, or use windowed Fourier transforms for a multi-ray model over localized windows.
4. Geometric Ray Tracing: Propagate these rays through the lens, employing analytic Snell’s Law Jacobians for full non-paraxial imaging.
5. Image Formation and PSF Assembly: Reconstruct spatially variant point spread functions (PSFs) and convolve with scene irradiance to generate differentiable images $G(x, y)$.
6. Optimization: Backpropagate gradients from image-space losses (MSE between $G$ and $F_\text{target}$, spot-diagram positional loss) to both metasurface and lens parameters.
7. Regularization: Enforce passive amplitude constraints ($A(r)\in [0,1]$), phase wrapping ($S(r)\in [0, 2\pi)$ via periodic activations), and minimum meta-atom feature sizes.

A summary table of learnable parameters:

Component	Main Parameters	Size/Count
Metalens	$\{w_0, w_1, w_2, w_3, h\}$	$10^3-10^5$
Phase/Amplitude	$S(r), A(r)$ via SIREN grid	$10^6$
Refractive Lens	$\{R_{front}, R_{back}, n, d\}$	$2$–$4$
Color Optics	$n_{eff}(\lambda)$ (per RCWA)	varies

2. MetaEOL Scaling Laws, Aberration Correction, and Empirical Performance

MetaEOL achieves simultaneous correction of spherical, comatic, and chromatic aberrations, demonstrated through co-optimized hybrid elements:

• Spherical/Non-Paraxial Aberration: For on-axis imaging, full-width half maximum (FWHM) of the PSF reduced by ~40% after joint optimization. Off-axis scenario with ±12° incidence exhibits a collapse of spot diagrams from ~100 μm to ~20 μm diameter and MSE image error reduction by a factor of 4–5.
• Chromatic Aberration: In color-corrected VR optics (single spherical element), co-optimized metalens equalizes focal planes of red, green, and blue channels to within ±10 μm at $z=7.2$ mm, raising the modulation transfer function (MTF) at 50 lp/mm from ~0.2 to ~0.7 (Zhu et al., 2023).

The framework’s differentiable architecture enables efficient GPU-accelerated scaling (e.g., 2 mm$^2$ metalens, $10^6$ grid points, $10^5$ rays per forward pass), with forward times on the order of 50 ms. These properties facilitate integration into compact imagers such as AR/VR headsets or smartphones, achieving performance superior to many-element classical lens stacks.

3. MetaEOL Meta-Task Prompting for High-Quality Unsupervised LLM Embeddings

MetaEOL in language modeling refers to "Meta-Task Prompting with Explicit One-Word Limitation," a method for unsupervised, fixed-size sentence embedding extraction from LLMs (e.g., LLAMA, Mistral) with zero parameter updating (Lei et al., 2024). The core innovation is to prompt the LLM with multiple diverse meta-task templates—Text Classification, Sentiment Analysis, Paraphrase Identification, Information Extraction—each formulated with an “in one word:” constraint. This forces the LLM to condense each meta-task’s semantic aspect into a single token, from which the d-dimensional embedding is extracted (last hidden state).

The process is formalized by:

Let $s$ denote an input sentence. For $T$ meta-tasks, each with $|\Omega_t|$ prompt templates $p_{t,i}(s)$, extract

$$h^{(t,i)}(s) = \text{LLM}_\theta(p_{t,i}(s))_{end} \in \mathbb{R}^d$$

Average over prompts and tasks:

$$m_t(s) = \frac{1}{|\Omega_t|} \sum_{i=1}^{|\Omega_t|} h^{(t,i)}(s), \quad e(s) = \frac{1}{T} \sum_{t=1}^T m_t(s)$$

where an optional $\ell_2$-normalization is used for cosine similarity tasks.

This architecture is empirically validated as follows:

• STS Benchmarks (Spearman ρ×100): On STS12–16 and SICK-R, MetaEOL (T=4 tasks, 2 prompts each) achieves 76–77 performance in LLMs (LLAMA2-7B, Mistral-7B, LLAMA3-8B), outperforming prior prompt-based (PromptEOL: 70–73), pooling (47–58), and competitive with unsupervised SimCSE-BERT (76).
• Ablations: Diversity of meta-task instruction, not mere prompt count, drives the improvement. Concatenation and max-pooling underperform compared to averaging. Adding meta-tasks monotonically increases representational quality.
• Scaling Law: Optimal layer for extraction scales as $\ell^* = \lfloor (1-\alpha)L \rfloor$ with $\alpha\approx0.10$, i.e., optimal within the last 10% of layers. For LLAMA2-70B with $L=80$, $\ell^*=72$ yields 78.06 STS, outperforming earlier choices.
• Transfer/Generalization: MetaEOL matches or exceeds fully supervised models on SentEval suite (MR, CR, etc.), with 91.81 average, beating larger trained encoders (ST5-Enc, 91.63).

4. Extended Depth of Focus Meta-Optics: Static MetaEOL Devices

Under the label MetaEOL, “Fast Extended Depth of Focus Meta-Optics for Varifocal Functionality” denotes a physical metasurface design that achieves extreme extension of the focal range via static phase coding (Whitehead et al., 2021). The device imposes a cubic phase profile,

$$\varphi(x, y) = \frac{2\pi}{\lambda}\left[ \sqrt{x^2 + y^2 + f^2} - f \right] + a(x^3 + y^3)$$

with cubic strength $a=1001\,\text{mm}^{-3}$ at $\lambda=633$ nm, to make the PSF depth-invariant over $\Delta z\simeq11$ mm, corresponding to 250× the traditional lens depth of focus (DOF) at f/1.75.

Key implementations:

• Meta-atom Geometry: Si₃N₄ pillars, 633 nm tall, on a 350 nm square grid, form a 2 mm-diameter aperture, producing $\sim$5.7 million scatterers with high transmission ($>$80%) and phase coverage (0–2π).
• Image Recovery: A single deconvolution kernel (PSF at central focus) suffices to reconstruct images over the range 3.5–14.5 mm, leveraging TV-regularized (total variation) deconvolution.
• Performance: Achieves 9.84 μm (50.8 cyc/mm) horizontal and 11.05 μm vertical line resolution (anisotropic due to cubic coding), NA up to 0.28, and >70% photon collection.
• Integration: Direct mounting onto commodity camera modules demonstrated varifocal imaging over 13–80 mm object distances, where comparable refractive lenses lost resolution outside native focus (Whitehead et al., 2021).

5. Comparative Tabulation of MetaEOL Paradigms

Name/Domain	Central Principle	Core Technical Distinction
Metalens-Enhanced Optical Lens	End-to-end differentiable wave–ray co-optimization	Gradient-based metalens + lens co-design (Zhu et al., 2023)
LLM Meta-Task Prompting	Averaged “one-word” meta-task LLM embedding	Multi-facet semantic compression—no tuning (Lei et al., 2024)
Extended DOF Meta-Optics	Cubic phase meta-optic for static EDOF/varifocality	Depth-invariant PSF, computational correction (Whitehead et al., 2021)

6. Practical Considerations, Limitations, and Future Directions

MetaEOL Hybrid Optics: Employs scalable GPU-accelerated PyTorch modules for wave-routings, SIREN surrogates, and high-dimensional ray-tracing. Design workflow includes pretraining surrogate solvers, initializing geometry, position-based (L_pos) and image-based (L_img) optimization, and integrating fabrication constraints. Scalability is maintained by efficient memory management and module chaining. Output phase patterns are directly exportable to e-beam lithography.

LLM Embeddings: The computational cost is dominated by eight LLM inference calls per sentence. No model fine-tuning or parameter updating is necessary. The gains saturate beyond four tasks/two prompts per task; prompt diversity, not repetition, is critical. Evaluation is currently restricted to English and sentence-level tasks, with diminishing returns for prompt/task count increases.

Extended DOF Optics: Metasurface fabrication is constrained by lithography resolution, aspect ratio, and transmission efficiency; pillar geometry is selected to balance phase coverage and etch logistics. Flatness and subwavelength period minimize unwanted diffraction, while PSF invariance supports general-purpose computational imaging.

Future work is expected to extend: (a) hybrid optics to multi-element stacks and miniaturized imaging, (b) LLM embedding schemes to multilingual and document-level use, and (c) meta-optic EDOF devices to integrated consumer imaging, biomedical optics, and autonomous vision.

7. Context and Broader Implications

The three MetaEOL instantiations represent convergent advancements in computational and physical engineering: integrated photonics via joint wave-ray numerical design (Zhu et al., 2023), unsupervised semantic representation without training overhead in LLMs (Lei et al., 2024), and static hardware-based varifocality in imaging (Whitehead et al., 2021). A plausible implication is acceleration in the computational design of compact, aberration-corrected cameras for both consumer and specialized technical domains, and for model-agnostic, resource-efficient neural language understanding. The MetaEOL approaches embody a trend toward hybridized, modular, and differentiable frameworks extending both the physical and algorithmic frontiers of imaging and representation systems.