# ⚛  L1 Principle — Second Harmonic Generation (SHG) Microscopy

**ID:** `L1-021` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Microscopy — *Chi-2 non-centrosymmetric NLO imaging*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D intensity
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554168

---

## 🧠 1. Introduction

**Second Harmonic Generation (SHG) Microscopy** is a **nonlinear inverse problem** whose unknown lives in **2D intensity** space, within the **Chi-2 non-centrosymmetric NLO imaging** sub-domain of **Microscopy**.

Measurements consist of photons collected by an optical detector via a **second harmonic nlo** sensing mechanism.

The forward operator applies, in order: L · excitation · nonlinear chi2 operator; ordered pixel-by-pixel sampling; detector accumulates flux over the exposure window.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D intensity is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 15); pulse_compression_drift dominates the stability cliff; sample_scattering and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

## ⚙ 2. Forward Model

Physical chain: **x** → L · excitation · nonlinear chi2 → Raster scan → Temporal integration → **y** (detector).

```
y = ∫_t dt S_raster `L.excitation.nonlinear_chi2` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.excitation.nonlinear_chi2` | L · excitation · nonlinear chi2 operator |
| `S.scan.raster` | Ordered pixel-by-pixel sampling |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Microscopy |
| Sub domain | Chi-2 non-centrosymmetric NLO imaging |
| Carrier | photon |
| Problem class | nonlinear_inverse |
| Solution space | 2D_intensity |
| Noise model | shot_poisson |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Existence of the recovered 2D intensity is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 15); pulse_compression_drift dominates the stability cliff; sample_scattering and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `shg` · **Forward operator:** `shg_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 512 |
| W | px | 512 |
| Na | — | 1 |
| Pixel nm | nm | 500 |
| Lambda ex nm | nm | 920 |
| Peak photons | photons | 100 |
| Pulse width fs | — | 100 |
| Sample scattering | — | 0 |
| Quasi phase mismatch | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 128 – 2048 |
| W | px | 128 – 2048 |
| Na | — | 0.6 – 1.1 |
| Pixel nm | nm | 200 – 1000 |
| Lambda ex nm | nm | 800 – 1100 |
| Peak photons | photons | 10 – 2000 |
| Pulse width fs | — | 50 – 500 |
| Sample scattering | — | 0.0 – 0.5 |
| Quasi phase mismatch | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 24.0

| Metric | Range |
|---|---|
| Psnr db | 12.0 – 42.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `300` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x27555a14a99c9b9b5d4eeadcf0a413c93b7e0ff95a6de4e24740c491919146aa`
- **Chain tx hash:** `0x835cc7a436d59b7909b3aed2559bed9119295e641e2885093301ba1efddbcd0f`
- **Chain block:** `41554168`

---

## File Mapping

This bundle consists of: `L1-021.md`, `L1-021.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-021.md` | Source of truth — edit this | Human or LLM |
| `L1-021.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 `L1-021`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._