# Zernike Trajectory Method for Elevation-Dependent Optimization

**Document Version**: 1.0  
**Created**: 2026-01-29  
**Author**: Mario (Clawdbot) + Antoine Letarte  
**Status**: Validated ✅

---

## Executive Summary

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:

1. **Tracks how each Zernike mode evolves** with elevation angle
2. **Fits a physics-based linear model** to the trajectory
3. **Provides integrated RMS metrics** for each aberration type
4. **Reveals which modes respond to axial vs lateral gravity loads**

**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.

---

## 1. Physics Background

### 1.1 Gravity Load Decomposition

At elevation angle θ from horizontal:

```
Axial load (along optical axis):   F_axial ∝ sin(θ)
Lateral load (perpendicular):      F_lateral ∝ cos(θ)
```

For a linear elastic structure, surface deformation is linear in these load components.

### 1.2 Zernike Coefficient Evolution

Each Zernike coefficient c_j follows:

```
c_j(θ) = a_j · (sin(θ) - sin(θ_ref)) + b_j · (cos(θ) - cos(θ_ref))
```

Where:
- `θ_ref` = reference angle (typically 20° for polishing/measurement)
- `a_j` = sensitivity of mode j to axial load change
- `b_j` = sensitivity of mode j to lateral load change

### 1.3 The Sensitivity Matrix

Define trajectory parameters:
```
τ(θ) = [sin(θ) - sin(θ_ref), cos(θ) - cos(θ_ref)]ᵀ
```

The full coefficient vector evolves as:
```
c⃗(θ) = S · τ(θ)
```

Where **S** is the sensitivity matrix (N_modes × 2).

---

## 2. Implementation

### 2.1 Extractor Location

```
optimization_engine/extractors/extract_zernike_trajectory.py
```

### 2.2 Basic Usage

```python
from optimization_engine.extractors.extract_zernike_trajectory import extract_zernike_trajectory

result = extract_zernike_trajectory(
    'path/to/solution.op2',
    reference_angle=20.0,  # Reference elevation (degrees)
)

# Mode-specific integrated RMS (nm)
print(result['coma_rms_nm'])
print(result['astigmatism_rms_nm'])
print(result['trefoil_rms_nm'])
print(result['spherical_rms_nm'])

# Total filtered RMS across all angles
print(result['total_filtered_rms_nm'])

# Linear model fit quality (should be > 0.95)
print(result['linear_fit_r2'])

# Sensitivity analysis
print(result['sensitivity_matrix'])
# {'coma': {'axial': 0.63, 'lateral': 36.04, 'total': 36.05}, ...}
```

### 2.3 Auto-Detection Features

The extractor automatically:
- Reads subcase labels from OP2 metadata (e.g., "20", "40", "60")
- Excludes manufacturing angle (90°) by default
- Sorts angles for proper trajectory fitting

### 2.4 Output Dictionary

```python
{
    # Mode-specific integrated RMS (nm)
    'coma_rms_nm': float,
    'astigmatism_rms_nm': float,
    'trefoil_rms_nm': float,
    'spherical_rms_nm': float,
    'secondary_astig_rms_nm': float,
    'secondary_coma_rms_nm': float,
    'quadrafoil_rms_nm': float,
    
    # Total filtered RMS (all modes, integrated)
    'total_filtered_rms_nm': float,
    
    # Model quality
    'linear_fit_r2': float,  # Should be > 0.95
    
    # Sensitivity analysis
    'sensitivity_matrix': {
        'coma': {'axial': float, 'lateral': float, 'total': float},
        'astigmatism': {...},
        ...
    },
    
    # Mode ranking (by sensitivity)
    'mode_ranking': ['spherical', 'coma', 'astigmatism', ...],
    'dominant_mode': str,
    
    # Metadata
    'angles_deg': [20.0, 30.0, 40.0, 50.0, 60.0],
    'reference_angle': 20.0,
}
```

---

## 3. Optimization Integration

### 3.1 AtomizerSpec Configuration

```yaml
extractors:
  - id: ext_trajectory
    name: "Zernike Trajectory"
    type: zernike_trajectory
    config:
      reference_angle: 20.0
      # Subcases auto-detected from OP2 labels

objectives:
  # Option A: Single combined objective
  - id: obj_total
    name: "Total Integrated RMS"
    source:
      extractor_id: ext_trajectory
      output_name: total_filtered_rms_nm
    direction: minimize
    weight: 1.0

  # Option B: Mode-specific objectives (multi-objective)
  - id: obj_coma
    name: "Coma RMS"
    source:
      extractor_id: ext_trajectory
      output_name: coma_rms_nm
    direction: minimize
    weight: 1.0

  - id: obj_astig
    name: "Astigmatism RMS"
    source:
      extractor_id: ext_trajectory
      output_name: astigmatism_rms_nm
    direction: minimize
    weight: 0.8
```

### 3.2 Recommended Optimization Strategy

**For SAT (Surrogate-Assisted Tuning):**
- Use `total_filtered_rms_nm` as primary objective
- Add mode-specific objectives as secondary for Pareto analysis
- SAT handles the expensive FEA evaluations efficiently

**For TPE (Tree-Parzen Estimator):**
- Good for single-objective optimization
- Use weighted combination if needed:
  ```
  J = w_coma * coma_rms + w_astig * astig_rms + w_total * total_rms
  ```

---

## 4. Validation Results (M1 Mirror, 2026-01-29)

### 4.1 Test Configuration
- **Model**: M1 GigaBIT mirror (1.2m Zerodur)
- **Angles**: 20°, 30°, 40°, 50°, 60° (5 subcases)
- **Reference**: 20° (polishing/measurement)

### 4.2 Linear Fit Quality
```
R² = 1.0000  ← PERFECT FIT
```

The physics model is **exactly correct** for this mirror design.

### 4.3 Mode-Specific Results

| Mode | Integrated RMS (nm) | Axial Sensitivity | Lateral Sensitivity |
|------|---------------------|-------------------|---------------------|
| Secondary Astig | 12.95 | 1.93 | 48.91 |
| Spherical | 10.57 | 19.84 | 68.08 |
| Coma | 9.14 | 0.63 | **36.04** |
| Trefoil | 6.83 | — | — |
| Astigmatism | 6.80 | — | — |

**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.

---

## 5. Troubleshooting

### 5.1 Low R² Value

If R² < 0.9:
- Check for contact nonlinearity (supports lifting off)
- Check for material nonlinearity
- Verify subcases have consistent boundary conditions

### 5.2 Missing Subcases

Ensure your FEA includes all required angles. Minimum recommended:
- 20° (reference)
- 40° (primary operational)
- 60° (secondary operational)

Adding 30° and 50° improves trajectory fit quality.

### 5.3 Subcase Label Detection

If auto-detection fails, specify manually:
```python
result = extract_zernike_trajectory(
    'solution.op2',
    subcases=[2, 5, 3, 6, 4],  # Subcase IDs
    angles=[20.0, 30.0, 40.0, 50.0, 60.0],  # Corresponding angles
)
```

---

## 6. References

- [ZERNIKE_OPD_METHOD.md](ZERNIKE_OPD_METHOD.md) — OPD correction for lateral displacement
- [ZERNIKE_FUNDAMENTALS.md](../ZERNIKE_FUNDAMENTALS.md) — Zernike polynomial basics
- AtomizerSpec v2.0 — Objective/extractor configuration

---

*Document maintained by Atomizer Framework. Last updated: 2026-01-29*