# ⚛  L1 Principle — Near-Field Scanning Optical Microscopy (NSOM/SNOM)

**ID:** `L1-164` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Scanning Probe — *Sub-diffraction optical microscopy via near-field aperture*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D intensity
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554243

---

## 🧠 1. Introduction

**Near-Field Scanning Optical Microscopy (NSOM/SNOM)** is a **linear inverse problem** whose unknown lives in **2D intensity** space, within the **Sub-diffraction optical microscopy via near-field aperture** sub-domain of **Scanning Probe**.

Measurements consist of photons collected by an optical detector via a **near field scanning optical** sensing mechanism.

The forward operator applies, in order: L · aperture near field operator; L · evanescent coupling operator; ordered pixel-by-pixel sampling; pixel-level spatial averaging on the detector.

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 holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 18); tip_sample_distance dominates the stability cliff; aperture_contamination and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

## ⚙ 2. Forward Model

Physical chain: **x** → L · aperture near field → L · evanescent coupling → Raster scan → Spatial integration → **y** (detector).

```
y = ∫_A dA S_raster `L.evanescent_coupling` `L.aperture_near_field` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.aperture_near_field` | L · aperture near field operator |
| `L.evanescent_coupling` | L · evanescent coupling operator |
| `S.scan.raster` | Ordered pixel-by-pixel sampling |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Scanning Probe |
| Sub domain | Sub-diffraction optical microscopy via near-field aperture |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | 2D_intensity |
| Noise model | shot_poisson |
| Integration axis | spatial |
| Difficulty delta | 5 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Existence of the recovered 2D intensity is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 18); tip_sample_distance dominates the stability cliff; aperture_contamination and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

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

## 📏 5. Operating Range (Ω)

**Center problem class:** `nsom` · **Forward operator:** `nsom_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Snr db | dB | 25 |
| Pixel nm | nm | 10 |
| Aperture nm | nm | 50 |
| Peak photons | photons | 500 |
| Optical coupling | — | 1 |
| Vibrational noise | — | 0 |
| Tip sample distance | — | 5 |
| Aperture contamination | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 64 |
| W | px | 64 |
| Snr db | dB | 0.0 – 40.0 |
| Pixel nm | nm | 1 – 100 |
| Aperture nm | nm | 10 – 500 |
| Peak photons | photons | 10 – 10000 |
| Optical coupling | — | 0.3 – 1.0 |
| Vibrational noise | — | 0.0 – 0.3 |
| Tip sample distance | — | 1 – 50 |
| Aperture contamination | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

| Metric | Range |
|---|---|
| Psnr db | 5.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x99d8c798110fcc57cd10d3c2a763abb60f2c31edd3dc4e1f80295042b154c2df`
- **Chain tx hash:** `0x46f6816e4e09df7fb3aa235f506bd68228aca9ca9f8ab6640fe53c8770914266`
- **Chain block:** `41554243`

---

## File Mapping

This bundle consists of: `L1-164.md`, `L1-164.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-164.md` | Source of truth — edit this | Human or LLM |
| `L1-164.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 `L1-164`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._