Hyperboloid Lenses

Along with the common spherical lens, OpticalRayTracer models another kind of lens, one with much better performance than the spherical, but one that is much more difficult to fabricate in glass.

When modeling a spherical lens, one need only specify one radius per sphere, as shown in the first graphic on this page. But when modeling a hyperboloidal curve, things are not so simple. Those properly trained in math will recall that circles, ellipses, parabolas and hyperbolas are all conic sections, or slices through a cone.

Examine the two graphic images to the right on this page. Click the top image repeatedly to cycle through the conic section images for a circle, ellipse, parabola and hyperbola. If you have a pair of anaglyphic eyeglasses (recommended), put them on and click the lower image in the same way for 3D views of the conic sections.

OpticalRayTracer relies on various derivative forms of the following equation to create/model hyperboloid lenses:

(12)

Where:

  • y = Vertical coordinate for a given x argument.
  • r = Lens sphere radius.
  • d = Lens depth, as above.
  • cf = Curvature factor, a lens shape fine-tuning prameter.

In OpticalRayTracer, the curvature factor has a meaningful range of 0 <= cf <= 0.25.

Equation (12), despite its central importance to the computations carried out, doesn't appear in the displayed form in OpticalRayTracer. It is instead the basis for a rather complex hyperboloid-line intersection routine used to process light paths during ray tracing.

Equation (12) is also restated as equation (13) below, used in OpticalRayTracer to draw hyperbolic lens profiles in the user interface:

(13)

Where:

  • x = Horizontal coordinate for a given y argument.
  • r = Lens sphere radius.
  • d = Lens depth, as above.
  • cf = Curvature factor, a lens shape fine-tuning prameter.

Here is an example lens design using the hyperboloid lens feature of OpticalRayTracer, similar to that provided in the help file. Start out with these settings:

In the "Configuration" tab:

  • Number of Tracing Beams = 6 (the default)
  • Diverging Beam Source = not selected

Create a lens with these traits:

  • Symmetrical = selected
  • Lens radius = 2 (the default)
  • IOR = 1.52 (the default)
  • X position = 0
  • Y position = 0
  • Left Sphere radius = 6 (applies to both radii)

Examine the lens focal point. The short focal length chosen for this lens produces a rather severe spherical aberration, as shown in the accompanying image. Now select "Left Hyerboloid," which, because you have enabled the "symmetrical" mode, will change both lens surfaces at once.

Now enter 0.074 for "Left Curv. Fact.". This will optimize the lens for this focal length and almost completely eliminate the spherical aberration that was evident before. Those who are following along without running OpticalRayTracer should click the accompanying graphic to see the change.