- Andrei Markin

# 3D Printed Optics 3: Off-Axis Parabolic Mirrors

Lens' are probably the most fundamental optical components, but mirrors can also act just like lens'. A particularly useful type is an off-axis parabolic mirror, this is a mirror which images a collimated beam to a single point at its focal length but crucially this image is perpendicular to the original axis.

These mirrors are extremely useful in optics as they are essentially achromatic and aspherical lens' which work in a large part of the spectrum (depending on what they are made of). But in general you might recognise the shape from something very common, satellite dishes.

The maths behind these mirrors is pretty straight forward, each ray in a parallel formation must hit the mirror at such an angle as to get reflected to a single point, defined as the focal point. The resulting curve is a parabola with the formula y = x^2 / 4f, where f is the parent focal length. An off-axis parabolic mirror can be thought of as just one section of this parabola:

To get a feel for these things I started by drawing a parabola in Matlab, the dimensions are in mm and I just played with the variables to see how it affects the shape of the curve and the resulting focal point (red x).

To generate a 3d file from this I could use the same script I used in my electron microscope project, but I think this is not necessary. 3D printing is far from perfect so I had an idea on how to do this much easier in Fusion 360. Start by drawing a cylinder with the dimensions of the mirror you would like:

Then simply import the generated parabola plot as an image and scale it so that the focal point is roughly where you want it, and that the curve intersects the entire centre of the cylinder:

In the example above the cylinder is 30mm in diameter which sets the reflective focal length to 30 mm. Now that we have the necessary cross-section we can trace it with the spline tool, then add a vertical line around the origin, enclosing the parabola. Then finally make a revolve-cut operation around the y axis at the origin to give the final shape.

And It's that simple, here is the final model of an off-axis parabolic mirror:

After it has been 3d printed the surface remains quite rough and not very reflective, especially in my case as I used black filament. But this can be improved by sanding the surface with fine sandpaper and then polishing with a mild abrasive (toothpaste works quite well). After polishing the surface we can use the weak specular reflection to judge how well the mirror performs:

It works!! surprisingly well actually, in this example I have a lamp around 30 cm away from the mirror and its image is focused to a spot around 2 mm in diameter. An interesting point is that the source actually gets focussed to two points, I'm still not exactly sure why this is but could be an artifact of the 3d printer. As the one this was printed on is a core xy type, the xy axes are separate from the z axes, and their calibration does not perfectly match so printed parts are slightly stretched in the x and y directions.

Even with these artifacts, if it's possible to make the surface reflective there should be quite a few applications. But after some early attempts this is not as simple as it seems so I will leave it for a future post.