From Classical Lens Design to Quantum Photonics via Differentiable Computing

The trajectory of the Eikonal formalism. The upper stream represents the evolution of classical optics from Hamilton (1828) to the 'Inflection Point' (2010). The lower stream represents the computational evolution. The 'Eikonal Bridge' connects these, utilizing the Bridge Identity to map classical characteristic functions into quantum operators.

This repository contains the companion materials for The Eikonal Bridge, a comprehensive technical book that unifies classical optical design with quantum photonics through the power of differentiable computing.

The Central Innovation: The Differentiable Eikonal Engine (DEE) framework uses JAX automatic differentiation to bridge two traditionally separate worlds—classical lens design and quantum photonics—through a single elegant identity:

φ_quantum = 2π × W_eikonal / λ

This bridge identity reveals that the classical eikonal function (optical path length) and quantum optical phase are the same mathematical object, enabling unified computational tools for both domains.

• 📚 12 Chapters in 4 Parts - Foundations → Computational Optics → Quantum Extensions → Production Practice
• 🔬 W/MN Duality Framework - Every topic from both forward analysis (Walther) and inverse design (Matsui-Nariai) perspectives
• 🎨 Production-Ready JAX Code - Complete, tested implementations with 100-1000× speedup
• ⚠️ Three-Axis Failure Framework - Systematic diagnostics for scalar eikonal limitations (P1-P5, M1-M5, T1-T3)
• 📊 Specification Translation Tables - Quantitative classical→quantum tolerance tightening factors (10-100×)
• 📝 80+ Paired Problems - W-type (analysis), MN-type (design), and Q-type (quantum) problem sets

• Optical Engineers transitioning from CODE V/Zemax to quantum photonics applications
• Quantum Physicists seeking practical lens design foundations for photonic systems
• Graduate Students in optics, photonics, and quantum information science
• Researchers exploring differentiable programming for optical design

Every chapter presents optical problems from dual perspectives:

This duality provides a universal problem-solving methodology that applies across classical and quantum domains.

When does the scalar eikonal approximation break down? Our systematic diagnostic framework identifies failure modes across three axes:

the-eikonal-bridge/
├── assets/
│   └── images/               # README and documentation images
│       └── eikonal_evolution_bridge.png
├── manuscript/               # LaTeX source files
│   ├── chapters/             # Individual chapter .tex files
│   ├── appendices/           # Appendix .tex files
│   ├── figures/              # Publication figures
│   └── main.tex              # Master document
├── code/                     # Python/JAX implementations
│   ├── dee_core/             # DEE framework core modules
│   ├── examples/             # Worked examples from each chapter
│   ├── problems/             # Problem solution code
│   └── figures/              # Figure generation scripts
├── solutions/                # Complete problem solutions manual
├── notebooks/                # Jupyter notebooks for exploration
├── data/                     # Sample datasets and results
└── docs/                     # Additional documentation

1. Clone or download the repository
2. Navigate to notebooks/ for interactive exploration
3. No heavy installation required for basic exploration!

# Clone the repository
git clone https://github.com/[username]/the-eikonal-bridge.git
cd the-eikonal-bridge

# Create virtual environment
python -m venv dee_env
source dee_env/bin/activate  # On Windows: dee_env\Scripts\activate

# Install dependencies
pip install -r requirements.txt

numpy>=1.21.0
matplotlib>=3.5.0
jax>=0.4.0
jaxlib>=0.4.0
scipy>=1.7.0
pyyaml>=6.0

import jax.numpy as jnp
from jax import grad

def eikonal_to_quantum_phase(W_eikonal, wavelength):
    """
    Bridge identity: phi_quantum = 2*pi * W_eikonal / lambda
    """
    return 2 * jnp.pi * W_eikonal / wavelength

# The gradient is automatically available
d_phase_dW = grad(eikonal_to_quantum_phase)

# Example: 1 wave of OPD at 1.55 um
W = 1.55e-6  # meters
wl = 1.55e-6  # meters
phase = eikonal_to_quantum_phase(W, wl)
print(f"Quantum phase: {phase:.4f} rad = {phase/(2*jnp.pi):.4f} waves")

A key feature of this book is quantitative guidance on tolerance tightening when transitioning from classical to quantum applications:

The key equation connecting waveguide coupling to quantum state evolution:

β_k = (dW_k/dz) × (2π/λ)

This identity enables unified design of classical CPO couplers and quantum photonic gates.

This work bridges decades of optical engineering wisdom with modern computational methods. Special thanks to:

• The optical design community for foundational knowledge
• The JAX development team for enabling differentiable computing

If you use this work in your research, please cite:

@book{lin2026eikonal,
  title     = {The Eikonal Bridge: From Classical Lens Design to 
               Quantum Photonics via Differentiable Computing},
  author    = {Lin, Jyh-Long},
  year      = {2025},
  publisher = {Open-source},
  series    = {},
  isbn      = {}
}

Contributions are welcome! Please see CONTRIBUTING.md for guidelines.

• 🐛 Bug Reports: Open an issue with a minimal reproducible example
• 💡 Feature Requests: Describe the use case and proposed solution
• 🔧 Code Contributions: Fork, create a feature branch, submit a PR

• Book Content: CC BY-NC-SA 4.0
• Code: MIT License

Current Version: 1.0 (Publication Preparation)
Last Updated: January 2026
Active Development: Yes

"The eikonal function is not just a mathematical convenience—it is the Rosetta Stone connecting classical ray optics to quantum wave mechanics."