geometries.toroidal

Toroidal Geometry

This module defines a toroidal surface geometry based on the Zemax definition: A base curve in the YZ plane (conic + polynomials) is rotated around an axis parallel to Y, offset by the radius of rotation R along Z.

z_y = yz_sag(y) = (c*y^2)/(1 + sqrt(1-(1+k)*c^2*y^2)) + sum(alpha_i * y^(2i+2)) z(x,y) = R - sqrt((R - z_y)^2 - x^2)

where: - R is the radius of rotation (X-Z curvature radius at vertex) - R_y = 1/c is the Y-Z curvature radius at vertex - k is the conic constant for the YZ curve - alpha_i are polynomial coefficients for the YZ curve (powers y^2, y^4, …)

Last Corrected by Manuel Fragata Mendes, July 2025

Classes

ToroidalGeometry(coordinate_system, ...[, ...])

Represents a simplified toroidal geometry (no Zernike terms as in zemax - - may be added later if necessary).

class ToroidalGeometry(coordinate_system: CoordinateSystem, radius_x: float, radius_y: float, conic: float = 0.0, coeffs_poly_y: list[float] = None, tol: float = 1e-10, max_iter: int = 100)

Represents a simplified toroidal geometry (no Zernike terms as in zemax - - may be added later if necessary).

Parameters:

• coordinate_system (CoordinateSystem) – The coordinate system.

• radius_x (float) – Radius of rotation R (X-Z radius).

• radius_y (float) – Base Y-Z radius R_y.

• conic (float, optional) – Conic constant k for the Y-Z curve. Defaults to 0.0.

• coeffs_poly_y (list[float], optional) – Polynomial coefficients alpha_i for the Y-Z curve, where coeffs_poly_y[i] corresponds to the coefficient for y^(2*(i+1)). Defaults to an empty list.

• tol (float, optional) – Tolerance for Newton-Raphson iteration. Defaults to 1e-10.

• max_iter (int, optional) – Maximum iterations for Newton-Raphson. Defaults to 100.

R_rot

Radius of rotation R (X-Z radius).

Type:

be.ndarray

R_yz

Base Y-Z radius R_y.

Type:

be.ndarray

k_yz

Conic constant k for the Y-Z curve.

Type:

be.ndarray

coeffs_poly_y

Polynomial coefficients alpha_i for the Y-Z curve.

Type:

be.ndarray

c_yz

Curvature of the Y-Z profile (1/R_yz).

Type:

be.ndarray or float

eps

Small epsilon value for safe division.

Type:

float

distance(rays)

Calculates the distance from the ray origin to the surface intersection using a robust Newton-Raphson method. This version uses the base conic intersection as a strong initial guess.

Parameters:

rays (RealRays) – The rays used for calculating distance.

Returns:

An array of propagation distances ‘t’ from each ray’s current position to its intersection point with the geometry.

Return type:

be.ndarray

flip()

Flip the geometry.

Changes the sign of the radius of rotation (R_rot) and the base Y-Z radius (R_yz). Updates the Y-Z curvature (c_yz) accordingly.

classmethod from_dict(data: dict) → ToroidalGeometry

Creates a ToroidalGeometry from a dictionary representation.

Parameters:

data (dict) – Dictionary containing toroidal geometry parameters.

Returns:

An instance of ToroidalGeometry.

Return type:

ToroidalGeometry

globalize(rays)

Convert rays from the local coordinate system to the global coordinate system.

Parameters:

rays (RealRays) – The rays to convert.

localize(rays)

Convert rays from the global coordinate system to the local coordinate system.

Parameters:

rays (RealRays) – The rays to convert.

sag(x: float, y: float) → float

Calculate the sag z(x, y) of the toroidal surface.

Parameters:

• x (float or be.ndarray) – X-coordinate(s).

• y (float or be.ndarray) – Y-coordinate(s).

Returns:

Sag value(s) of the toroidal surface.

Return type:

float or be.ndarray

scale(scale_factor: float)

Scale the geometry parameters.

Parameters:

scale_factor (float) – The factor by which to scale the geometry.

set_radius(value: float) → None

Set the radius of curvature.

Parameters:

value (float) – The new radius of curvature.

surface_normal(rays)

Calculates the surface normal of the geometry at the given rays.

Parameters:

rays (RealRays) – The rays, positioned at the surface, for which to calculate the surface normal.

Returns:

The surface normal components (nx, ny, nz).

Return type:

tuple[be.ndarray, be.ndarray, be.ndarray]

to_dict() → dict

Converts the geometry to a dictionary.

Returns:

A dictionary representation of the toroidal geometry.

Return type:

dict

update_normalization(semi_aperture: float) → None

Update the normalization attributes of the geometry based on its defined normalization_mode (‘auto’ or ‘manual’). Base geometry generally does not maintain a normalization radius.

Parameters:

semi_aperture (float) – The current semi-aperture of the surface.