270 lines
6.8 KiB
Markdown
270 lines
6.8 KiB
Markdown
# Zernike Trajectory Method for Elevation-Dependent Optimization
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**Document Version**: 1.0
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**Created**: 2026-01-29
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**Author**: Mario (Clawdbot) + Antoine Letarte
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**Status**: Validated ✅
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---
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## Executive Summary
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The Zernike Trajectory Method provides **mode-specific optimization objectives** for telescope mirrors operating across multiple elevation angles. Instead of optimizing a weighted sum of discrete WFE values, this method:
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1. **Tracks how each Zernike mode evolves** with elevation angle
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2. **Fits a physics-based linear model** to the trajectory
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3. **Provides integrated RMS metrics** for each aberration type
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4. **Reveals which modes respond to axial vs lateral gravity loads**
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**Key result**: For a well-designed support system, the linear model achieves R² ≈ 1.0, meaning deformation is entirely predictable from the gravity load decomposition.
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---
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## 1. Physics Background
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### 1.1 Gravity Load Decomposition
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At elevation angle θ from horizontal:
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```
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Axial load (along optical axis): F_axial ∝ sin(θ)
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Lateral load (perpendicular): F_lateral ∝ cos(θ)
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```
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For a linear elastic structure, surface deformation is linear in these load components.
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### 1.2 Zernike Coefficient Evolution
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Each Zernike coefficient c_j follows:
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```
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c_j(θ) = a_j · (sin(θ) - sin(θ_ref)) + b_j · (cos(θ) - cos(θ_ref))
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```
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Where:
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- `θ_ref` = reference angle (typically 20° for polishing/measurement)
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- `a_j` = sensitivity of mode j to axial load change
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- `b_j` = sensitivity of mode j to lateral load change
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### 1.3 The Sensitivity Matrix
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Define trajectory parameters:
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```
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τ(θ) = [sin(θ) - sin(θ_ref), cos(θ) - cos(θ_ref)]ᵀ
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```
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The full coefficient vector evolves as:
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```
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c⃗(θ) = S · τ(θ)
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```
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Where **S** is the sensitivity matrix (N_modes × 2).
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---
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## 2. Implementation
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### 2.1 Extractor Location
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```
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optimization_engine/extractors/extract_zernike_trajectory.py
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```
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### 2.2 Basic Usage
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```python
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from optimization_engine.extractors.extract_zernike_trajectory import extract_zernike_trajectory
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result = extract_zernike_trajectory(
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'path/to/solution.op2',
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reference_angle=20.0, # Reference elevation (degrees)
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)
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# Mode-specific integrated RMS (nm)
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print(result['coma_rms_nm'])
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print(result['astigmatism_rms_nm'])
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print(result['trefoil_rms_nm'])
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print(result['spherical_rms_nm'])
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# Total filtered RMS across all angles
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print(result['total_filtered_rms_nm'])
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# Linear model fit quality (should be > 0.95)
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print(result['linear_fit_r2'])
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# Sensitivity analysis
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print(result['sensitivity_matrix'])
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# {'coma': {'axial': 0.63, 'lateral': 36.04, 'total': 36.05}, ...}
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```
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### 2.3 Auto-Detection Features
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The extractor automatically:
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- Reads subcase labels from OP2 metadata (e.g., "20", "40", "60")
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- Excludes manufacturing angle (90°) by default
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- Sorts angles for proper trajectory fitting
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### 2.4 Output Dictionary
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```python
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{
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# Mode-specific integrated RMS (nm)
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'coma_rms_nm': float,
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'astigmatism_rms_nm': float,
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'trefoil_rms_nm': float,
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'spherical_rms_nm': float,
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'secondary_astig_rms_nm': float,
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'secondary_coma_rms_nm': float,
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'quadrafoil_rms_nm': float,
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# Total filtered RMS (all modes, integrated)
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'total_filtered_rms_nm': float,
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# Model quality
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'linear_fit_r2': float, # Should be > 0.95
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# Sensitivity analysis
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'sensitivity_matrix': {
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'coma': {'axial': float, 'lateral': float, 'total': float},
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'astigmatism': {...},
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...
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},
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# Mode ranking (by sensitivity)
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'mode_ranking': ['spherical', 'coma', 'astigmatism', ...],
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'dominant_mode': str,
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# Metadata
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'angles_deg': [20.0, 30.0, 40.0, 50.0, 60.0],
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'reference_angle': 20.0,
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}
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```
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---
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## 3. Optimization Integration
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### 3.1 AtomizerSpec Configuration
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```yaml
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extractors:
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- id: ext_trajectory
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name: "Zernike Trajectory"
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type: zernike_trajectory
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config:
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reference_angle: 20.0
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# Subcases auto-detected from OP2 labels
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objectives:
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# Option A: Single combined objective
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- id: obj_total
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name: "Total Integrated RMS"
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source:
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extractor_id: ext_trajectory
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output_name: total_filtered_rms_nm
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direction: minimize
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weight: 1.0
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# Option B: Mode-specific objectives (multi-objective)
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- id: obj_coma
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name: "Coma RMS"
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source:
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extractor_id: ext_trajectory
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output_name: coma_rms_nm
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direction: minimize
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weight: 1.0
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- id: obj_astig
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name: "Astigmatism RMS"
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source:
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extractor_id: ext_trajectory
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output_name: astigmatism_rms_nm
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direction: minimize
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weight: 0.8
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```
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### 3.2 Recommended Optimization Strategy
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**For SAT (Surrogate-Assisted Tuning):**
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- Use `total_filtered_rms_nm` as primary objective
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- Add mode-specific objectives as secondary for Pareto analysis
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- SAT handles the expensive FEA evaluations efficiently
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**For TPE (Tree-Parzen Estimator):**
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- Good for single-objective optimization
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- Use weighted combination if needed:
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```
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J = w_coma * coma_rms + w_astig * astig_rms + w_total * total_rms
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```
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---
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## 4. Validation Results (M1 Mirror, 2026-01-29)
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### 4.1 Test Configuration
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- **Model**: M1 GigaBIT mirror (1.2m Zerodur)
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- **Angles**: 20°, 30°, 40°, 50°, 60° (5 subcases)
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- **Reference**: 20° (polishing/measurement)
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### 4.2 Linear Fit Quality
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```
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R² = 1.0000 ← PERFECT FIT
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```
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The physics model is **exactly correct** for this mirror design.
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### 4.3 Mode-Specific Results
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| Mode | Integrated RMS (nm) | Axial Sensitivity | Lateral Sensitivity |
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|------|---------------------|-------------------|---------------------|
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| Secondary Astig | 12.95 | 1.93 | 48.91 |
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| Spherical | 10.57 | 19.84 | 68.08 |
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| Coma | 9.14 | 0.63 | **36.04** |
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| Trefoil | 6.83 | — | — |
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| Astigmatism | 6.80 | — | — |
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**Key insight**: Coma is almost entirely driven by lateral loads (axial/lateral ratio = 0.02). Optimizing lateral support locations will have the biggest impact on coma.
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---
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## 5. Troubleshooting
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### 5.1 Low R² Value
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If R² < 0.9:
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- Check for contact nonlinearity (supports lifting off)
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- Check for material nonlinearity
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- Verify subcases have consistent boundary conditions
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### 5.2 Missing Subcases
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Ensure your FEA includes all required angles. Minimum recommended:
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- 20° (reference)
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- 40° (primary operational)
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- 60° (secondary operational)
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Adding 30° and 50° improves trajectory fit quality.
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### 5.3 Subcase Label Detection
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If auto-detection fails, specify manually:
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```python
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result = extract_zernike_trajectory(
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'solution.op2',
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subcases=[2, 5, 3, 6, 4], # Subcase IDs
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angles=[20.0, 30.0, 40.0, 50.0, 60.0], # Corresponding angles
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)
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```
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---
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## 6. References
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- [ZERNIKE_OPD_METHOD.md](ZERNIKE_OPD_METHOD.md) — OPD correction for lateral displacement
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- [ZERNIKE_FUNDAMENTALS.md](../ZERNIKE_FUNDAMENTALS.md) — Zernike polynomial basics
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- AtomizerSpec v2.0 — Objective/extractor configuration
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---
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*Document maintained by Atomizer Framework. Last updated: 2026-01-29*
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