# ⚛  L1 Principle — Thin Film Interference — transfer-matrix optical design

**ID:** `L1-245` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Layered-stack optical transfer-matrix*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** stack thickness and n
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553989

---

## 🧠 1. Introduction

**Thin Film Interference — transfer-matrix optical design** is a **linear inverse problem** whose unknown lives in **stack thickness and n** space, within the **Layered-stack optical transfer-matrix** sub-domain of **Electromagnetics**.

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

The forward operator applies, in order: L · transfer matrix operator; L · layer phase operator; L · boundary match operator; integration over a band of frequencies.

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

## ⚙ 2. Forward Model

Physical chain: **x** → L · transfer matrix → L · layer phase → L · boundary match → Frequency integration → **y** (detector).

```
y = ∫dω `L.boundary_match` `L.layer_phase` `L.transfer_matrix` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.transfer_matrix` | L · transfer matrix operator |
| `L.layer_phase` | L · layer phase operator |
| `L.boundary_match` | L · boundary match operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Layered-stack optical transfer-matrix |
| Carrier | em |
| Problem class | linear_inverse |
| Solution space | stack_thickness_and_n |
| Noise model | gaussian |
| Integration axis | frequency |
| 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:** `thin_film` · **Forward operator:** `thin_film_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 40 |
| N error | — | 0 |
| N layers | — | 5 |
| N wavelengths | — | 401 |
| Substrate error | — | 0 |
| Thickness error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 10 – 60 |
| N error | — | 0.0 – 0.05 |
| N layers | — | 1 – 100 |
| N wavelengths | — | 11 – 10001 |
| Substrate error | — | 0.0 – 0.05 |
| Thickness error | — | 0.0 – 0.05 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 30.0

| 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:** `0xf8bc8ce7dd2a17826058979d41d0a62a7f1ea3668c55dd8e8ffba835184eb08e`
- **Chain tx hash:** `0xa6622612b5410b7f67d069163195c3b096076b8c5602c1daddeaeccf759c61ff`
- **Chain block:** `41553989`

---

## File Mapping

This bundle consists of: `L1-245.md`, `L1-245.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-245.md` | Source of truth — edit this | Human or LLM |
| `L1-245.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 `L1-245`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._