# ⚛  L1 Principle — Photonic Inverse Design — nanostructure topology optimization

**ID:** `L1-248` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electromagnetics — *Adjoint-based photonic topology optimization*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** binary permittivity map
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553990

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## 🧠 1. Introduction

**Photonic Inverse Design — nanostructure topology optimization** is a **nonlinear inverse problem** whose unknown lives in **binary permittivity map** space, within the **Adjoint-based photonic topology optimization** sub-domain of **Electromagnetics**.

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

The forward operator applies, in order: L · fdfd operator; L · adjoint operator; L · density filter 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 · fdfd → L · adjoint → L · density filter → Frequency integration → **y** (detector).

```
y = ∫dω `L.density_filter` `L.adjoint` `L.fdfd` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.fdfd` | L · fdfd operator |
| `L.adjoint` | L · adjoint operator |
| `L.density_filter` | L · density filter operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electromagnetics |
| Sub domain | Adjoint-based photonic topology optimization |
| Carrier | em |
| Problem class | nonlinear_inverse |
| Solution space | binary_permittivity_map |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 5 |
| L dag | 3.5 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | FOM_rel_score |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `photonic_adjoint` · **Forward operator:** `photonic_adjoint_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| N iter | — | 500 |
| Snr db | dB | 40 |
| N freqs | — | 16 |
| Source error | — | 0 |
| Material error | — | 0 |
| Fab discretization error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 2048 |
| W | px | 32 – 2048 |
| N iter | — | 10 – 100000 |
| Snr db | dB | 10 – 60 |
| N freqs | — | 1 – 256 |
| Source error | — | 0.0 – 0.1 |
| Material error | — | 0.0 – 0.1 |
| Fab discretization error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.8

| Metric | Range |
|---|---|
| Fom | 0.1 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **FOM_rel_score**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x9c59cc2b5427c124d57676e4130cb6869bb667bdf5b9849d80f586d7f6fb1f81`
- **Chain tx hash:** `0x84c8def73f7af19fb1446e102a15a03611dda727073b8709eb077f7e8241f763`
- **Chain block:** `41553990`

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## File Mapping

This bundle consists of: `L1-248.md`, `L1-248.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-248.md` | Source of truth — edit this | Human or LLM |
| `L1-248.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 `L1-248`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._