Solid Glass Kaleidoscopes: Total Internal Reflection Without Mirrors

4 days ago

7 min read

By Rocco (Roi) Ramon - licensed architect, founder of Studio Yabaye

A kaleidoscope is normally a hollow instrument: two or more mirrors enclose a wedge of air, and the light travels through that air between reflections. There is a second way to build it. The wedge can be a solid block of glass, with no mirror coating at all. The light travels inside the glass and is reflected by the polished faces of the block itself.

Sir David Brewster described this form in "A Treatise on the Kaleidoscope" (Edinburgh, 1819), in Chapter XII, titled "On Kaleidoscopes in which the effect is produced by total reflection from the interior surfaces of transparent solids". This article follows that chapter and compares a solid prism with a front-surface mirror system. Chapter numbers are those of the 1819 edition. In the 1858 second edition the same subject is Chapter XIV.

What total internal reflection is

Light changes direction when it passes from one transparent material into another. Passing from glass into air, a ray bends away from the normal - the line perpendicular to the surface.

At one particular angle of incidence the outgoing ray would run exactly along the surface. This is the critical angle. For any ray that meets the surface more obliquely than that, there is no outgoing ray. The light is turned back into the glass and obeys the ordinary law of reflection. This is total internal reflection.

The critical angle depends on the refractive index of the material. For glass with an index of 1.5 in air, the sine of the critical angle is 1 / 1.5, which gives an angle of about 42 degrees from the normal. The condition holds only where the outside of the reflecting face is a medium of lower refractive index, in practice air.

Why the reflection loses almost no light

A metal surface absorbs part of the light at every reflection. Brewster opens Chapter XII with this point: on the most perfectly polished metals a very considerable quantity of light is absorbed, and even at the greatest obliquities there is a manifest difference in intensity between the direct and the reflected beam.

Total internal reflection has no absorbing metal layer. Brewster's statement is that in total reflection from the second surfaces of transparent bodies "the loss of light is very inconsiderable", and that the reflection is more brilliant than that of polished metal.

The difference matters in a kaleidoscope because the reflections repeat: a sector on the far side of the field has been reflected several times.

Brewster's solid kaleidoscope

The block is cut as a wedge, like the air space of a simple two-mirror kaleidoscope.

The two reflecting faces. They are inclined at an angle that is an even aliquot part of a circle - the same rule as for mirrors. Brewster requires them to be "ground perfectly flat and highly polished".
The junction. The edge where the two faces meet must be made as fine as possible.
The third face. The upper surface, opposite the junction, is rough ground.
The two ends. The object end and the eye end are parallel and well polished.

If the glass is colourless and good, Brewster states, the eye sees the same appearance as in the simple kaleidoscope.

Why the angle condition is met: Brewster does not give this reasoning, but it follows from the law of refraction. For glass of index 1.5 and a flat end face perpendicular to the axis, a ray inside the glass is inclined at most about 42 degrees to the axis. It therefore meets a side face at about 48 degrees or more from the normal, beyond the critical angle.

The difficulties Brewster lists

Brewster names two defects of the solid form, one requirement for the material and one complication of focus.

Loss of light inside the glass. The light crosses the full length of the block, and Brewster notes that the glass is not perfectly transparent. He considers this loss counterbalanced by the intensity of the totally reflected light.

The junction of the two faces. He lists "the difficulty of obtaining a perfect junction of the two reflecting planes". In a solid block the edge is produced by grinding two faces until they meet, so a chipped or rounded edge stays in the block. Brewster adds that this defect does not exist when the instrument is intended to give rectilineal or annular patterns.

Homogeneity of the glass. The block must be "a piece of glass entirely free from veins". A vein, a bubble or a tint in the glass lies in the light path.

Focus. Brewster notes that an eye able to see objects distinctly at five inches will not see them distinctly at the end of a glass prism five inches long. He attributes this to the glass making the rays diverge from a point nearer than the object itself. His solution is a prism only two or three inches long, with a lens at the eye end whose focal length is less than the length of the prism. The eye end of the glass can itself be ground to a spherical form to serve as the lens.

One advantage is stated at the end of the chapter. The solid form is, in Brewster's words, peculiarly fitted for polycentral instruments, where three faces reflect. The face that would otherwise be left rough is polished, and the prism is cut to the angles given in Chapter XI of the 1819 edition.

Solid prism vs front-surface mirror system

A front-surface mirror carries its reflective coating on the face the light reaches first, so the light does not pass through the glass. The comparison with a solid prism:

Loss per reflection. Solid prism: close to none at the reflecting face. Front-surface mirror: the share absorbed by the coating.
Loss between reflections. Solid prism: absorption along the full length of the glass. Mirror system: the path is air.
The reflecting surface. Solid prism: bare polished glass on the outside of the block. Total reflection depends on air behind that face, so anything in optical contact with it - a fingerprint, adhesive, a mount - can let light leak out at that spot. Mirror system: the reflecting faces are inside the tube.

The solid prism moves the loss from the reflecting surface to the body of the glass. The instruments I build at Studio Yabaye use front-surface mirrors. This article describes the solid form from Brewster's text and from general optics.

What to check when looking at a prism kaleidoscope

"Prism kaleidoscope" is used loosely. It can mean a solid glass block that reflects by total internal reflection, or a hollow instrument whose mirrors are arranged as a triangular prism. Ask which one it is. The checklist below covers the solid form.

General buying criteria are in what to look for when buying a kaleidoscope, and the instrument families are listed in types of kaleidoscopes.

Checklist

The description states whether the instrument is a solid block or a hollow mirror system.
The reflecting faces of a solid block show no contact marks.
The glass is colourless and free of veins and bubbles.
The junction edge is fine and unchipped.
The object end is in focus from the eye end.
The image was judged under direct, strong light.

FAQ

What is a solid glass kaleidoscope?

It is a kaleidoscope in which the wedge between the reflectors is a solid block of glass and not air. The two long faces of the block are ground flat and polished, and they reflect the light from the inside by total internal reflection. No metal mirror coating is used. Brewster described it in 1819.

What is total internal reflection?

It is the complete reflection of light at the inside surface of a transparent material. It occurs when light travelling in glass meets a glass-to-air boundary at an angle from the normal larger than the critical angle. No ray passes out through the surface, and the reflected ray follows the ordinary law of reflection.

What is the critical angle of glass?

For glass with a refractive index of 1.5 in air, the critical angle is about 42 degrees, measured from the line perpendicular to the surface. It is found from the relation sine of the critical angle = 1 / refractive index. Glass with a higher refractive index has a smaller critical angle.

Is a prism kaleidoscope brighter than a mirror kaleidoscope?

At the reflecting faces, yes: Brewster wrote in 1819 that total reflection loses very little light, while polished metal absorbs a considerable quantity. But the light also crosses the full length of the glass block, which is not perfectly transparent. Brewster names that loss as one of two defects of the solid form.

Did Brewster make a solid glass kaleidoscope?

Chapter XII of the 1819 Treatise gives a full construction: a wedge of vein-free glass, two flat polished faces at an even aliquot part of a circle, a rough third face and polished parallel ends. The chapter is written as an instruction. It does not record whether or how many such instruments were made.

Where else is total internal reflection used?

Porro prism binoculars fold the light path with right-angle glass prisms whose faces reflect at about 45 degrees, beyond the critical angle of common optical glass. Optical fibres guide light along a glass core by repeated total reflection at the boundary with a cladding of lower refractive index than the core.

About Studio Yabaye

I build Studio Yabaye kaleidoscopes and teleidoscopes with front-surface mirror systems in several configurations, housed in a brass tube, cut, fitted and hand-finished in the studio. The studio is based in Israel and ships worldwide. The current range is at all products, and questions can be sent through the contact page.

Reference: Sir David Brewster, "A Treatise on the Kaleidoscope", Edinburgh, 1819 (public domain), Chapter XII.