12. Geometry Overview

Geometries define the shapes of optical surfaces and play a key role in ray tracing by handling two critical operations:

  1. Ray-Surface Intersection: Determining the intersection point (or propagation distance) between a ray and the surface.

  2. Surface Normal Calculation: Computing the surface normal at the intersection point, which is essential for applying Snell’s law or the law of reflection.

12.1. Key Components

A geometry is defined by:

  • Coordinate System: Each geometry relies on a CoordinateSystem object, which specifies:

    • Position: x, y, z (surface origin in the global coordinate space)

    • Rotation: rx, ry, rz (Euler angles or equivalent rotation representation)

    • Reference Coordinate System: Enables nested or relative transformations.

  • Geometry-Specific Parameters: Additional attributes tailored to the surface type. For example:

  • StandardGeometry: This class in standard.py handles spherical and conic surfaces.

  • Aspheric Geometries: Implemented in even_asphere.py and odd_asphere.py. - Custom Geometries: User-defined parameters. This flexibility allows for a wide range of surface shapes, including freeforms.

Note

Geometries are defined in their local coordinate systems. They are then transformed based on their CoordinateSystem argument, which specifies their position and orientation in the global coordinate space.

12.2. Intersection and Normal Computation

Geometries provide methods for:

  1. Finding the Intersection Point:

    • For simple shapes, this is computed using closed-form equations.

    • For complex shapes, iterative methods like Newton-Raphson are used to find the intersection.

    • The NewtonRaphsonGeometry class serves as the base for geometries requiring iterative solutions.

  2. Computing Surface Normals:

    • Normals are derived from the mathematical description of the geometry and are essential for determining the ray’s direction after refraction or reflection.

12.3. Supported Geometry Types

Optiland includes a wide range of built-in geometries:

  • Standard: Handles spherical and conic surfaces.

  • Planes: Flat surfaces with infinite or finite extent.

  • EvenAsphere: Described by polynomial terms for deviations from a sphere.

  • OddAsphere: Similar to even aspheres but with additional terms for odd powers.

  • Biconic: A surface with different radii and conic constants in x and y.

  • Toroidal: Defined by two radii of curvature, allowing for toroidal shapes.

  • PlaneGrating and StandardGrating: Surfaces with diffraction gratings.

  • Polynomial and Chebyshev: Useful for advanced freeform optical systems.

  • Zernike Surfaces: Represented by Zernike polynomials. For a detailed mathematical description of the Zernike geometry, see the Zernike Geometry Mathematics Reference.

  • Forbes Surfaces: As described in the corresponding Forbes Surface gallery example, these surfaces are defined following the convention by the papers: [1] G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express 19(10), 9923–9941 (2011) and [2] G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20, 2483-2499 (2012). The qpoly.py module in optilandgeometriesforbes was adapted from the implementation in the prysm <https://github.com/brandondube/prysm?tab=readme-ov-file> package.

  • NURBS: Non-Uniform Rational B-Splines for highly flexible freeform surfaces. See the NURBS Freeform Optics gallery example (Freeform Optics) for usage.

  • Custom Geometries: Users can easily extend the framework by subclassing the BaseGeometry (analytical geometries) or NewtonRaphsonGeometry (iterative geometries) classes.

12.4. Extensibility

Adding new geometries is straightforward:

  1. Subclass the BaseGeometry base class or the NewtonRaphsonGeometry class.

  2. For BaseGeometry implement the distance(rays), sag(x, y) and surface_normal(rays) methods. For NewtonRaphsonGeometry implement the sag(x, y) and _surface_normal(x, y) methods.

  3. Optionally, define additional parameters in the constructor for the geometry’s specific shape.

Tip

See the Surface Overview section for how geometries integrate with surfaces.