# ⚛  L1 Principle — Fresnel Diffraction — paraxial wave propagation

**ID:** `L1-241` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Paraxial (Fresnel) diffraction integral*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex 2D field
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553988

---

## 🧠 1. Introduction

**Fresnel Diffraction — paraxial wave propagation** is a **linear inverse problem** whose unknown lives in **complex 2D field** space, within the **Paraxial (Fresnel) diffraction integral** sub-domain of **Electromagnetics**.

Measurements consist of electromagnetic field measurements via a **intensity camera** sensing mechanism.

The forward operator applies, in order: L · paraxial propagate operator; L · fresnel convolve operator; L · intensity capture operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · paraxial propagate → L · fresnel convolve → L · intensity capture → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.intensity_capture` `L.fresnel_convolve` `L.paraxial_propagate` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.paraxial_propagate` | L · paraxial propagate operator |
| `L.fresnel_convolve` | L · fresnel convolve operator |
| `L.intensity_capture` | L · intensity capture operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Paraxial (Fresnel) diffraction integral |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | complex_2D_field |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `fresnel` · **Forward operator:** `fresnel_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Z m | m | 1 |
| Snr db | dB | 30 |
| Lambda nm | nm | 633 |
| Z dist error | — | 0 |
| Aperture error | — | 0 |
| Wavelength error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 – 4096 |
| W | px | 64 – 4096 |
| Z m | m | 0.001 – 1000 |
| Snr db | dB | 0 – 40 |
| Lambda nm | nm | 1 – 10000 |
| Z dist error | — | 0.0 – 0.05 |
| Aperture error | — | 0.0 – 0.1 |
| Wavelength error | — | 0.0 – 0.02 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 26.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x962286a8a0878837f180eb1ef49a26fe3456eebd93ac875ffe3719696c0450a3`
- **Chain tx hash:** `0x400affcdc98972afee3a3617b38912e3d4e9ebff2a355d392f5c059b51da9e51`
- **Chain block:** `41553988`

---

## File Mapping

This bundle consists of: `L1-241.md`, `L1-241.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-241.md` | Source of truth — edit this | Human or LLM |
| `L1-241.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-241`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._