# ⚛  L1 Principle — Digital Holography (off-axis and in-line intensity holograms)

**ID:** `L1-071` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Coherent Imaging — *Interferometric complex-field recovery*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** complex 2D
> **📡 Carrier:** photon · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554169

---

## 🧠 1. Introduction

**Digital Holography (off-axis and in-line intensity holograms)** is a **linear inverse problem** whose unknown lives in **complex 2D** space, within the **Interferometric complex-field recovery** sub-domain of **Coherent Imaging**.

Measurements consist of photons collected by an optical detector via a **interferometric** sensing mechanism.

The forward operator applies, in order: L · modulation · reference operator; L · add coherent operator; phase is lost; only intensity is measured; pixel-level spatial averaging on the detector.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Off-axis single-shot: ill-posed without spatial bandwidth separation (tilt must exceed twice max object spatial frequency). In-line phase-shifting: well-posed with N >= 3 phase steps; stability dominated by reference-phase calibration error and wavelength drift.

## ⚙ 2. Forward Model

Physical chain: **x** → L · modulation · reference → L · add coherent → Magnitude-squared → Spatial integration → **y** (detector).

```
y = ∫_A dA |·|² `L.add_coherent` `L.modulation.reference` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.modulation.reference` | L · modulation · reference operator |
| `L.add_coherent` | L · add coherent operator |
| `L.modulus_squared` | Phase is lost; only intensity is measured |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Coherent Imaging |
| Sub domain | Interferometric complex-field recovery |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | complex_2D |
| Noise model | shot_poisson |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Off-axis single-shot: ill-posed without spatial bandwidth separation (tilt must exceed twice max object spatial frequency). In-line phase-shifting: well-posed with N >= 3 phase steps; stability dominated by reference-phase calibration error and wavelength drift.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | phase_L2_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `complex_field_recovery` · **Forward operator:** `off_axis_digital_hologram`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1024 |
| W | px | 1024 |
| Z mm | mm | 100 |
| Photon count | — | 1000 |
| Wavelength nm | nm | 632.8 |
| Tilt angle mrad | — | 10 |
| Wavelength drift | — | 0 |
| Reference tilt error | — | 0 |
| Propagation distance error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 – 4096 |
| W | px | 256 – 4096 |
| Z mm | mm | 1.0 – 500.0 |
| Photon count | — | 10 – 50000 |
| Wavelength nm | nm | 400 – 1550 |
| N phase shifts | — | 3 – 8 |
| Tilt angle mrad | — | 1.0 – 50.0 |
| Wavelength drift | — | 0.0 – 0.002 |
| Phase shift error | — | 0.0 – 0.1 |
| Reference tilt error | — | 0.0 – 2.0 |
| Propagation distance error | — | 0.0 – 5.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 32.0 dB PSNR

| Metric | Range |
|---|---|
| Psnr db | 18.0 – 42.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `1200` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xd93b970467e3ac7a98b365b06ed1fe861772605ba4604d3789d07ce52bdcc821`
- **Chain tx hash:** `0xa034295c686077a8ba8f7cf6230b4614efbf784b9d7edad609f8501e5a632d68`
- **Chain block:** `41554169`

---

## File Mapping

This bundle consists of: `L1-071.md`, `L1-071.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-071.md` | Source of truth — edit this | Human or LLM |
| `L1-071.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 `L1-071`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._