Optics 006 · Ray worlds, boundaries, and natural optics

Floating Image Mirascope

Where do two facing parabolic mirrors form the floating image you can see but not touch? At the hole in the top mirror. Face two identical parabolic mirrors toward each other so that each focus sits at the center of the other mirror, and put the object at the bottom center. The top mirror turns its light into a parallel beam, and the bottom mirror focuses that beam back up at the opening, where a real image hangs in midair.

A close cutaway of a mirascope labels the two paraboloids, each with its focus at the other's vertex, along with the central opening, the hidden point object and the glowing real-image point. Path segments in different colors make the two reflections, the collimated light between them, aperture clipping and the crossing in free space easy to follow.

Captured solid angle

\eta_{\Omega}

$0.50$

Real-image height

h_{\mathrm{image}}

$50\%$

Optical-path spread

\Delta\mathrm{OPL}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Floating Image Mirascope

BackgroundA mirascope is two identical parabolic mirrors facing each other, with the focus of each mirror at the vertex of the other. Each surface follows $z=\frac{r^2}{4f}$ . Light from the hidden object reflects off the upper mirror into a parallel beam, and the lower mirror then focuses that beam into a real image at the opening.

Why it mattersWhere do two facing parabolic mirrors form the floating image you can see but not touch?

Start with the essentials

Focus question: Where do two facing parabolic mirrors form the floating image you can see but not touch?
One-sentence intuition: In this ideal geometry, every accepted ray has the same optical path length: $\Delta\mathrm{OPL}=0$ . Clipping the aperture changes the collected solid angle and the brightness, but not where the image forms.

Core mathematical model

Paraboloid profile

z=\frac{r^2}{4f}

The drawn mirror surface and the normals used for reflection come from the same focal length.

Equal optical path length

\mathrm{OPL}=\sum_j n_j\ell_j,\qquad \Delta\mathrm{OPL}=0

In the ideal air-filled geometry, every accepted ray travels the same total distance from object to image.

Collected solid-angle fraction

\eta_{\Omega}=\frac{\Omega_{\mathrm{accepted}}}{2\pi}

This is the fraction of the emitting hemisphere that the aperture captures. Directions outside it never reach the image.

Common difficulties

Treating the floating image as a hologram

Typical misconceptionThe floating object is a screen effect, or a virtual image that exists only from one viewpoint.

Better mental modelThe rays really do converge in open air above the opening, so the image is real. Moving your viewpoint changes which of those rays enter the eye, not where they meet.

Run the experiment

01
Scene 1: Open mirascope
Follow one color sequence from the hidden object to the upper mirror, down the collimated middle segment, and from the lower mirror to the free-space image point.
What to observe: Every accepted ray reflects once off the upper mirror and once off the lower mirror, then passes through the same real-image point.
02
Scene 2: Aperture clipping
Reduce the outer mirror aperture. Notice that it is separate from the central opening, which has its own label.
What to observe: Clipping removes the high-angle rays and shrinks the collected solid angle. The image height is still set by the focal length.
03
Scene 3: Walk-around parallax
In the walk-around scene, orbit the camera without changing the optical geometry.
What to observe: As the camera orbits, the real image stays put. Only the set of rays you can see and the parallax change.