Surfaces

Torch Lens Maker is designed to support many surfaces, and be easily extensible with custom surface models.

Axially symmetric surfaces:

• Disk: A circular planar surface
• SphereByRadius: Spherical arc parameterized by radius of curvature R
• SphereByCurvature: Spherical arc parameterized by curvature $C=1/R$
• Parabola: Parabolic arc parameterized by parabolic coefficient $A$.
• Conic: Arc of a conic, parameterized by curvature $C$ and conic constant $K$.
• Asphere: Asphere parameterized by a conic model + asperic coefficient vector.

Freeform surfaces:

• XYPolynomial: XY Polynomial model, parameterized as a conic + coefficient matrix.

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.

Disk

import torchlensmaker as tlm

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

tlm.show2d(optics, end=5)
tlm.show3d(optics, end=5)

SphereByCurvature

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 planar disk), 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 SphereByRadius class which uses radius parameterization and polar distance functions. In fact SphereByCurvature cannot represent an exact half circle (R = D/2) due to the gradient becoming infinite.

import torchlensmaker as tlm

surface = tlm.SphereByCurvature(diameter=10, C=1/-25)

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

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

SphereByRadius

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. You will also not be able to change the sign of the radius during optimization.

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.SphereByRadius(diameter=10, R=5)
surface2 = tlm.SphereByRadius(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 principal 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, C=1/-15, K=-1.6, alphas=[0.00012])

optics = tlm.Sequential(
    tlm.PointSourceAtInfinity(25),
    tlm.Gap(5),
    tlm.RefractiveSurface(surface, materials=("air", "water")),
)

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