Surfaces

Torch Lens Maker is designed to support many surfaces. Currently supported are:

• CircularPlane: A circular planar surface (aka a disk)
• Sphere: with curvature parameterization
• SphereR: Sphere with radius parameterization
• Parabola: A parabola on the X axis $x=a{r}^2$
• Asphere: The asphere model often used in optics

Work in progress

More surface types coming soon, hopefully 😁 I also want to document how to add custom surfaces easily, as a lot of work as gone into that while designing the library. Basically any sag function $x=g(r)$ can be added, or even any implicit surface described by $F(x,y,z)=0$, not necessarily axially symmetric.

CircularPlane

import torchlensmaker as tlm

surface = tlm.CircularPlane(diameter=4.0)
optics = tlm.Sequential(
    tlm.PointSource(10),
    tlm.Gap(5),
    tlm.Turn((45, 0)),
    tlm.ReflectiveSurface(surface),
)

tlm.show2d(optics, end=5)

Sphere

A section of a sphere, parameterized by signed curvature. Curvature is the inverse of radius: C = 1/R.

This parameterization is useful because it enables clean representation of an infinite radius section of sphere (which is really a plane), and also enables changing the sign of C during optimization.

In 2D, this surface is an arc of circle. In 3D, this surface is a section of a sphere (wikipedia calls it a "spherical cap")

For high curvature arcs (close to a half circle), it's better to use the SphereR class which uses radius parameterization and polar distance functions. In fact this class cannot represent an exact half circle (R = D/2) due to the gradient becoming infinite, use SphereR instead.

import torchlensmaker as tlm

surface = tlm.Sphere(diameter=10, R=-25)

optics = tlm.Sequential(
    tlm.PointSourceAtInfinity(beam_diameter=5),
    tlm.Gap(5),
    tlm.Turn((20, 0)),
    tlm.ReflectiveSurface(surface),
)

tlm.show2d(optics, end=12)
tlm.show3d(optics, end=12)

SphereR

A section of a sphere, parameterized by signed radius.

This parameterization is useful to represent high curvature sections including a complete half-sphere. However it's poorly suited to represent low curvature sections that are closer to a planar surface.

In 2D, this surface is an arc of circle. In 3D, this surface is a section of a sphere (wikipedia call it a "spherical cap")

import torchlensmaker as tlm

surface1 = tlm.SphereR(diameter=10, R=5)
surface2 = tlm.SphereR(diameter=10, R=-5)

optics = tlm.Sequential(
    tlm.ReflectiveSurface(surface1),
    tlm.Gap(10),
    tlm.ReflectiveSurface(surface2),
)

tlm.show2d(optics, end=12)
tlm.show3d(optics, end=12)

Parabola

A parabolic surface on the X axis: $X=A{R}^2$.

import torchlensmaker as tlm

surface = tlm.Parabola(diameter=10, A=0-.03)

optics = tlm.Sequential(
    tlm.PointSource(35),
    tlm.Gap(-1/(4*surface.A.item())),
    tlm.ReflectiveSurface(surface),
)

tlm.show2d(optics, end=4)
tlm.show3d(optics, end=4)

Asphere

The typical Aphere model, with a few changes:

• X is the principal optical axis in torchlensmaker
• Internally, the radius parameter R is represented as the curvature $C=\frac{1}{R}$, to allow changing the sign of curvature during optimization
• k is the conic constant

$$X(r)=\frac{C{r}^2}{1+\sqrt{1-(1+K)r^2{C}^2}}+{\alpha }_4{r}^4+{\alpha }_6{r}^6\dots$$

In 2D, the derivative with respect to r is:

$${\mathrm{\nabla }}_{r}X(r)=\frac{Cr}{\sqrt{1-(1+K)r^2{C}^2}}+4{\alpha }_4{r}^3+6{\alpha }_6{r}^5\dots$$

In the 3D rotationally symmetric case, we have $r^2=y^2+z^2$.

The derivative with respect to y (or z, by symmetry) is:

$$F_y^{\prime }(x,y,z)=\frac{Cy}{\sqrt{1-(1+K)(y^2+z^2)C^2}}+y(4{\alpha }_4(y^2+z^2)+6{\alpha }_6(y^2+z^2{)}^2+\dots )$$

import torchlensmaker as tlm


surface = tlm.Asphere(diameter=30, R=-15, K=-1.6, A4=0.00012)

optics = tlm.Sequential(
    tlm.PointSourceAtInfinity(25),
    tlm.Gap(5),
    tlm.RefractiveSurface(surface, material="water-nd"),
)

tlm.show2d(optics, end=10)
tlm.show3d(optics, end=10)