The window a diamond is cut inside

The angle past which light cannot leave finds diamond’s critical angle, 24.4 degrees, and notes in passing that “the brilliant cut is an arrangement of facet angles designed to make that sequence” — light entering, bouncing internally and leaving through the top — happen. The sentence is where the subject of this essay begins. Designing that arrangement is a problem with a precise answer, found by a mining engineer’s son in 1919, and the answer says as much about the material as about the cut.

The round brilliant is the familiar shape: a flat top, the table; a ring of sloping crown facets around it; the widest point, the girdle; and below it a cone of pavilion facets meeting at a point, the culet. Light that enters the table travels downward into the pavilion. If the pavilion faces were silvered, getting that light back out of the top would be a matter of aiming. They are not. They reflect only because the light meets them from inside at more than the critical angle, and at the last step it must meet the top at less than the critical angle or it will be reflected back down. The design is a window between conditions that pull in opposite directions.

Light turned back twice #

Marcel Tolkowsky’s proportions come from a small book, Diamond Design, which he wrote in 1919 at twenty-two while studying engineering in London, having grown up in a family of Antwerp diamond cutters. He treated the stone as a problem in geometric optics, traced rays through a cross-section as the drawing does, and derived angles for the pavilion and the crown that returned the most light while spreading it into colours. His pavilion angle, 40.75 degrees, and his crown angle, 34.5, remain the reference for the “ideal cut” — modern computer ray-tracing of full three-dimensional stones has refined them but not overturned them.

The drawing follows six rays straight down through the table. Each one does the same thing. The first pavilion face it meets is inclined at 40.75 degrees to the horizontal, so the ray meets it at 40.75 degrees from its normal, well past the critical angle; it is totally reflected and sent almost horizontally across the stone. It meets the opposite face at 57.75 degrees and is reflected again, now upward. It reaches the top at 17 degrees from the normal, inside the critical angle, and leaves.

Two total reflections that turn a ray back on itself are the same trick as the corner that sends light home, where three faces at right angles reverse every ray exactly. The brilliant is a looser version, in cross-section a pair of faces at roughly 98 degrees to each other rather than 90, and deliberately so: an exact retroreflector would send the light straight back towards the light source, which is the observer’s head, and a stone that returned the viewer’s own shadow would look dark. The small departure from a right angle spreads the returned light into a cone that the eye can see.

Three conditions and the window they leave #

Each of the three angles in the ray’s journey is a simple function of the pavilion angle $\theta_p$, found by reflecting the ray’s direction in each face in turn: $\theta_p$ at the first face, $180^\circ - 3\theta_p$ at the second and $|4\theta_p - 180^\circ|$ back at the top. The first two must exceed the critical angle, so that the ray is reflected; the third must be less, so that it gets out. The drawing plots all three against the dashed critical angle.

The first condition is easy. Any pavilion steeper than 24.4 degrees reflects a vertical ray at the first face. The second fails only for very deep pavilions. The third is the one that decides: the ray reaches the top at the critical angle when $\theta_p$ is 38.9 or 51.1 degrees, and only between them does it escape. Outside the window the two reflections still work — but they send the ray back to the top too obliquely, it is reflected back down into the stone, and it wanders until it finds a way out, almost always through the pavilion.

A window twelve degrees wide sounds generous, but the calculation so far is for light entering straight down, and a stone is lit from every direction. Light entering the table at an angle is refracted towards the normal — diamond’s high index bends even grazing light to within 24.4 degrees of vertical — but that residual tilt is added to or subtracted from every subsequent angle, and it narrows the window from both sides. That is what the fourth drawing takes into account.

Too shallow, just right, too deep #

The three stones make the window visible. The middle one is Tolkowsky’s. The one on the left has a pavilion six degrees too shallow, and its rays come back up to the top at 44 degrees, far past the critical angle; they are reflected down again, bounce off the pavilion at angles no longer suited to returning them, and half of them leave through the bottom. The one on the right is eleven degrees too deep, and its rays meet the top at 28 degrees, just outside the critical angle — and every one of them is lost, after a longer journey down into the narrow tip of the stone.

Seen from above, light that leaves through the pavilion is light missing from the face of the stone. A shallow stone shows a glassy, washed-out ring where the reflection of the girdle should be — cutters call it a fish-eye — and a deep one shows a dark centre, a nail-head. Both are cut that way for the same commercial reason: a rough crystal yields more weight when the cut follows its shape rather than the optimum, and a diamond’s price rises faster than its weight. The optical penalty of keeping weight is exactly the leakage the drawing shows.

How much comes back, for four materials #

The fourth drawing replaces the single straight-down ray with light arriving from every direction within 40 degrees of vertical and entering anywhere across the table, and counts the fraction that comes back out of the top. It also varies the material. The result is the most important fact about gemstones and optics: brilliance is set by the refractive index before it is set by the cut.

A higher index shrinks the critical angle, and a smaller critical angle loosens every condition at once. The pavilion reflections succeed over a wider range, and — this is less obvious — the exit condition, which needs the ray to reach the top within the critical angle, is helped too, because a high index bends incoming light more steeply towards the vertical and so feeds the pavilion with rays closer to the ideal straight-down one. Diamond returns nearly nine-tenths of the light over a range of several degrees around 42. Cubic zirconia, index 2.15, does almost as well at a slightly steeper angle. Sapphire at 1.77 returns less than half at its best. Glass, at 1.52 and with a critical angle of 41 degrees, cannot hold light by total reflection in this geometry at all: most of what enters it leaves through the back, whatever angle it is cut to, and a glass “diamond” relies on a mirrored foil behind it to sparkle.

Going beyond diamond does not keep helping. Synthetic moissanite, silicon carbide, has an index of about 2.65 and a smaller critical angle still, and it returns light as well as diamond does; but it is doubly refracting, and looking down through its table shows every pavilion facet edge doubled, a blur a trained eye spots at once. Diamond’s cubic crystal structure gives it one index in every direction, so its facet edges stay single and sharp. The best material for the window is not only the one with the highest index but the one with a single index.

The opposite problem is solved by the same arithmetic. A light-emitting diode is a small block of semiconductor with an index near 3.5 and a critical angle of 17 degrees, and the light it makes inside must leave rather than return. The cone light has to find to get out counts how little of it does through a flat face, and the remedies — roughened surfaces, domed lenses, tilted facets — are gem cutting run in reverse, shaping a high-index body so that as much internal light as possible meets its surface inside the critical angle rather than outside it.

The colours a stone throws out #

Brilliance is white light returned. Fire is white light returned in pieces: the flashes of spectral colour that a diamond throws when it or the viewer moves. It comes from dispersion — the index of diamond is 2.408 for red light and 2.451 for violet — but the dispersion alone is modest, and the drawing shows why a cut can multiply it.

Light leaving a face squarely is bent very little, and red and violet leave almost together: half a degree apart for diamond at ten degrees from the normal. Light leaving a face near the critical angle is bent a great deal, skimming out almost along the surface, and there a small change in index swings the outgoing direction through a large angle — the derivative of the refraction angle diverges at the critical angle. A degree short of it, diamond’s red and violet leave more than three degrees apart, and a cut that sends light out of the crown facets near their critical angle spreads each white ray into a visible spectrum. This is the same amplification that puts the rainbow at its angle and the halo at twenty-two degrees: an extremum or a near-grazing angle turns a small dispersion into a visible separation.

Brilliance and fire trade against each other through the crown. A larger table lets in more light and returns more of it as brilliance; steeper crown facets and a smaller table send more of it out obliquely and spread more of it into colour. Tolkowsky’s 53 per cent table and 34.5-degree crown are one balance between them. Cubic zirconia, whose dispersion is about half as large again as diamond’s, throws more fire from the same cut, which is one of the ways a gemmologist’s eye tells the two apart: the imitation is, if anything, too colourful.

Five centuries of opening the window #

Diamonds were not always bright. The earliest cut stones, in the fourteenth century, were natural octahedral crystals with their faces polished — point cuts — whose faces meet at about 55 degrees to the crystal’s axis, well outside the window, and which return little light and look dark and oily. The table cut ground off one point of the octahedron to make a flat top; the rose cut of the sixteenth and seventeenth centuries covered a flat-bottomed dome with triangular facets and returned light by surface reflection more than by internal reflection. None of these shapes was designed around the critical angle, because nobody yet thought of a gem as an optical instrument.

The brilliant’s ancestors — the Mazarin and Peruzzi cuts of the seventeenth century and the old mine cut of the eighteenth — added a pavilion of facets beneath the girdle, and with it the double internal reflection. Their pavilions were deep and their tables small, and their proportions were set by the rough crystal and by the cutters’ eyes rather than by calculation. The steam-driven bruting machine of the 1870s, which rounded a stone’s girdle mechanically, made the round shape cheap, and within a generation the round brilliant had displaced everything else. Tolkowsky’s contribution was to take the shape the tools had made possible and ask what angles it should have. The history of diamond cutting is the history of cutters finding the window by trial long before anybody wrote down the inequalities that define it.

The critical angle that gemmologists measure with #

The same angle that makes a diamond bright is the gemmologist’s main instrument for telling stones apart. A gem refractometer is a small hemisphere of high-index glass with the stone laid flat on its top, wetted with a drop of dense liquid to make optical contact. Light sent up through the glass is totally reflected wherever it meets the stone at more than the critical angle for glass against gem, and transmitted where it meets it at less, so the field of view is divided into a bright part and a dark part by a sharp line — the boundary at the angle where light can no longer leave — and a scale in the eyepiece reads the stone’s index off the line’s position. Quartz reads 1.544 and 1.553, sapphire 1.762 and 1.770, the pairs being the two indices of a crystal that splits light by polarisation.

The method has a ceiling, and diamond is above it. The contact liquid must have a higher index than the stone, and the densest practical liquid stops at about 1.81; above that, no boundary line appears. A stone that shows no reading on a refractometer is therefore one of the few materials with an index above 1.81, and diamond, cubic zirconia and moissanite are told apart by other means — thermal conductivity, double refraction, weight. The high index that opens diamond’s window is also what puts it beyond the instrument built on the same physics.

Where the cross-section stops #

Two dimensions. Every ray here travels in a single plane through the stone’s axis. A real brilliant has fifty-seven or fifty-eight facets arranged with eight-fold symmetry, and most rays travel on skew paths that never lie in any one plane; full three-dimensional ray tracing gives light-return figures somewhat different from these and a range of good pavilion angles rather than a single best one. The window’s existence and its dependence on the index survive; its exact edges do not.

Perfect material and polish. The trace assumes a flawless, perfectly polished stone with no absorption. Inclusions scatter light out of the designed paths, and a slightly rounded facet edge — the usual result of wear — sends light out where it should have been reflected.

Geometric optics. The facets are millimetres across, thousands of wavelengths, so rays are a fair description. Total reflection is not quite total at the scale of a wavelength, where the reflected beam is displaced along the face, but the displacement is far too small to matter here.

What the pictures cannot show #

The drawings show where light goes and not what a stone looks like, and the second depends on the first in a way no ray trace captures. A diamond’s appearance is made of the moving pattern of bright and dark facets — scintillation — as the stone, the lights or the viewer’s head move, and the eye’s response to that pattern is much of what people value. It depends on the size and distribution of the light sources, on the viewer’s distance and on the reflection of the viewer’s own head, which appears in every well-cut stone as dark facets and which a stone lit by a single overhead spotlight shows most strongly. Grading laboratories now photograph stones under standardised lighting to measure it, and the measurement is still argued about.

Still open: what a best cut means #

Tolkowsky optimised for a single figure of merit, and he could compute only a two-dimensional section. Modern cut grading uses three-dimensional ray tracing of the actual stone, measured by a scanner to a few micrometres, and asks it for three separate quantities — brightness, fire and scintillation — that no cut maximises together. Which combination the eye prefers is not a question optics can settle; it has been studied with panels of observers comparing stones side by side, and the answers depend on the lighting and, to a degree that makes the question genuinely hard, on the observer. Whether there is a single best cut, or only a family of cuts each best for a different light and a different eye, remains open.

The habit worth carrying away is to ask of any device built from total internal reflection which boundary must reflect and which must transmit. A stone is held between two uses of the same critical angle — faces that must exceed it and a face that must not — and the window between them opens as the index rises. Cutting cannot open a window the material has closed, which is why no amount of craft makes glass behave like diamond, and why an LED, needing the opposite, is shaped by the same arithmetic run backwards.