Optical Design & Simulation (Prisms)

Optical design, analytical modeling, and 3D ray-tracing simulations for the laser prism scanner known as Hexastorm.

This package provides the optical calculations used by:

• Hexastorm Design: github.com/hstarmans/hexastorm_design
• Hackaday Project: hackaday.io/project/21933-open-hardware-transparent-polygon-scanner
• RepRap Project: reprap.org/wiki/Open_hardware_fast_high_resolution_LASER
• Video Walkthrough: YouTube Explanation

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Optical Layout & Ray Tracing

Layout Overview (Shown at 38° Start-of-Scan Angle)

• Clockwise Prism Rotation & Start-of-Scan (SOS): The transparent N-BK7 polygon prism rotates clockwise, sweeping the refracted beam across the scan field from left to right. Synchronization must occur at the beginning of each scanline, which is why the pick-off mirror is positioned on the left (X = -11 mm).
• Mounting Wall Clearance: The pick-off mirror is positioned at Y = 25 mm, ensuring the reflected beam travels at Y = 27.92 mm—passing cleanly in front of Cylindrical Lens 2 (Y ≥ 29.5 mm) and the laserhead mounting wall to reach the photodiode sensor on the far right (X = 35 mm).
• Spot Circularization & Anamorphic Correction: Two crossed cylindrical lenses (CL1 at Y = -29 mm and CL2 at Y = 31 mm) circularize the elliptical laser diode output and suppress cross-scan error caused by facet-to-datum manufacturing tolerances.
• Telecentric Projection: Over the active exposure range (X ∈ [-5.2, +5.2] mm), the beam exits at a perpendicular 90° angle, maintaining uniform spot geometry and enabling seamless multi-lane stitching.

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Key Optical Advantages

1. Telecentric Exposure: Unlike reflective galvanometer or polygonal mirror scanners where the incident angle varies dynamically across the scanline, refraction through rotating parallel faces produces a naturally telecentric scan (always 90° to the substrate). This eliminates parallax distortion when stitching adjacent lanes.
2. 4× Less Sensitive to Facet Errors: For a refractive prism with refractive index n ≈ 1.5, a 1° facet-to-datum manufacturing imperfection causes only ~0.5° of beam deflection ((n - 1) × α). In contrast, a reflective mirror doubles angular error to 2.0° (2 × α).
3. Scalable Multi-Head Stacking (90° Tilt Optimization): Multi-beam single-polygon scanners must tilt the polygon axis (e.g., α ≤ 45° relative to substrate motion) to prevent simultaneous beam lines from overlapping, which reduces the effective scan angle and usable scan length (y_length = sin(α) × S_L). Hexastorm eliminates this limitation by keeping each modular head at the optimal 90° static polygonal tilt angle (100% scan length utilization, sin(90°) = 1.0). Wide substrate coverage is achieved by modularly tiling independent single-beam heads across width and depth (staggered array), providing seamless lane stitching without line-overlap constraints.
4. Open Hardware Prior Art: An unpatented optical architecture based on foundational prior art by Lindberg (US Patent 3,253,498, 1966), establishing robust open-source freedom of use.

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Features

1. Analytical Model (prisms.analytical)

   • Implements James C. Wyant's optical testing and aberration theory for plane-parallel rotating plates.
   • Computes diffraction-limited Gaussian beam waist spot size and Rayleigh range.
   • Calculates longitudinal and transversal focus shift through rotating polygon facets.
   • Computes 3rd-order Seidel aberrations (spherical, coma, astigmatism), λ OPD RMS, and Strehl ratio.
   • Analyzes scanline duty cycle, non-uniform sweep velocity, and cross-scan facet-to-datum errors.

2. Ray-Tracing Simulation (prisms.system & prisms.library)

   • Non-sequential 3D ray tracing using upstream pyOpTools.
   • Models N-BK7 polygon prisms, Edmund Optics cylindrical lenses, fold mirrors, and photodiode detection targets.
   • Automatically determines synchronization photodiode hit angles (find_object('diode')) and focal plane positioning.

3. Live CAD Optics Verification (prisms.cad_verifier)

   • Connects to a running FreeCAD session live via XML-RPC (port 9875 / FreeCAD MCP).
   • Automatically extracts global transforms of optical components (lenstube, CLens1, prism, CLens2, mirror, photodiode).
   • Validates mechanical alignment against optical tolerances (laser axis alignment, cylinder lens centering, confocal focal plane distance).
   • Pushes exact 3D ray compounds (405 nm violet laser) back into the active FreeCAD document (Simulation/Rays).
   • Run with:

     uv run python -m prisms.cad_verifier

4. Modern Interactive Notebooks (Notebooks/)

   • Marimo Reactive Apps: Launch interactive simulations with live sliders via uv run marimo edit Notebooks/plot_system.py or uv run marimo edit Notebooks/system_compact.py.
   • Universal Plotly 3D Views: Hardware-accelerated WebGL visualization of rays and optical components.

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Installation & Setup

This repository uses uv for fast, reproducible, and modern dependency management.

Clone the repository and synchronize the environment:

# Sync core dependencies and dev/notebook groups
uv sync --all-groups

To install this package in editable mode in an external project (such as hexastorm_design):

uv add --editable /path/to/opticaldesign

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Running Tests

Run the complete automated test suite with pytest:

uv run pytest -v

Lint and format code using ruff:

uv run ruff check .
uv run ruff format .

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Interactive Notebooks

Launch either reactive Marimo application with live sliders:

# Full scanner simulation (with cylindrical lenses)
uv run marimo edit Notebooks/plot_system.py

# Compact layout & Fresnel reflection analysis
uv run marimo edit Notebooks/system_compact.py

Or run directly in headless/script mode:

uv run python Notebooks/plot_system.py
uv run python Notebooks/system_compact.py

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Mathematical Summary (Wyant Aberration Theory)

The analytical model in prisms.analytical implements the wavefront aberration theory published by James C. Wyant in Basic Wavefront Aberration Theory for Optical Metrology.

1. 3rd-Order Seidel Aberrations for Tilted Plane-Parallel Plate

A rotating polygon facet of thickness T and refractive index n tilted at angle of incidence θ introduces 3rd-order wavefront aberrations:

• Spherical Aberration (Wyant p. 42, eq. 72): sabr = -T / f_numb⁴ × ((n² - 1) / (128 × n³))
• Coma (Wyant p. 44, eq. 75): coma = -T × θ / f_numb³ × ((n² - 1) / (16 × n³)) × cos(φ)
• Astigmatism (Wyant p. 45, eq. 77): astig = -T × θ² / f_numb² × ((n² - 1) / (8 × n³)) × cos²(φ)

2. Optical Path Difference (OPD) & Strehl Ratio

• Wavefront Error Variance (Wyant p. 37, eq. 62): Evaluated across the normalized circular pupil (ρ ∈ [0, 1], φ ∈ [0, 2π]) via double quadrature: σ² = (1/π) × ∫∫ [ΔW(ρ, φ)]² ρ dρ dφ - [(1/π) × ∫∫ ΔW(ρ, φ) ρ dρ dφ]²
• λ OPD RMS: λ_RMS = σ / λ
• Strehl Ratio (Wyant p. 39, eq. 67): Strehl ≈ 1 - (2π × λ_RMS)² + (2π × λ_RMS)⁴ / 2

3. Literature Benchmark

The numerical integration in p.lambda_opd_rms(θ) reproduces the published Wyant literature values (tested in tests/test_analytical.py for T = 35 mm, f = 90 mm, D = 1.2 mm, n = 1.53, λ = 405 nm):

Tilt Angle (θ)	Wyant Literature (λ OPD RMS)	prisms.analytical (λ OPD RMS)
10°	0.005	0.0055
24°	0.032	0.0315
30°	0.050	0.0493

4. Scanner Geometry & Focus Shifts

• Gaussian Waist Radius: waist = (2 × λ / π) × f_numb
• Rayleigh Length: z_r = π × waist² / λ
• Longitudinal Focus Shift: slong = ((n - 1) / n) × T (Wyant p. 41, eq. 68)
• Transversal Focus Shift: disp = T × sin(θ) × (1 - cos(θ) / sqrt(n² - sin²(θ))) (Wyant p. 41, eq. 70)
• Duty Cycle: duty_cycle = max_recommended_angle / max_angle_incidence
• Cross-Scan Error: cross_err = max(tan((n - 1) × apex_angle) × focal_distance, transversal_shift(apex_angle))

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Architecture

opticaldesign/
├── src/prisms/
│   ├── __init__.py       # Top-level exports (PrismProperties, PrismScanner, Polygon)
│   ├── analytical.py     # Wyant physics formulas, aberrations, and Strehl calculations
│   ├── library.py        # Custom pyOpTools components (regular Polygon prism)
│   ├── system.py         # Complete optical system layout and ray-tracing routines
│   └── viewer.py         # 3D Plotly visualization integration
├── Notebooks/
│   ├── plot_system.py    # Interactive Marimo scanner simulation (cylinder layout)
│   └── system_compact.py # Interactive Marimo compact layout & reflection analysis
├── docs/
│   └── images/
│       └── prism_scanner_38deg.png # 3D ray-tracing layout diagram
├── tests/
│   ├── test_analytical.py # Analytical formulas & Wyant literature benchmark tests
│   ├── test_library.py    # Polygon geometry tests
│   ├── test_system.py     # Ray propagation & diode hit detection tests
│   └── test_viewer.py     # 3D Plotly visualization tests
├── pyproject.toml         # Standard PEP 621 configuration managed by uv
└── uv.lock                # Fully pinned, reproducible lockfile

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License

GPL-3.0-or-later.