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Getting Rid of Those Bow Tie Blues - Part I
by Bob Keller

Many faceters are familiar with bow ties. High fashion neckwear excepted, bow ties are generally not a good thing in the faceter's world. Of course I'm referring to the "bow tie effect" in faceted gemstones, which pertains to undesirable dark regions in a stone caused by light leaks in their pavilions. A commonly encountered pattern of bright and dark regions in stones suffering from this malady consists of opposed pairs of triangular regions sharing a common, central vertex.

The underlying causes responsible for the bow tie effect stem from fundamental optical properties of gem materials and design problems that buck harder as the aspect ratio (L/W) of gemstones increase. When these causes are understood and appreciated the faceter is better equipped to judge which gemstone designs are likely to suffer from bow ties, and why cutting them in materials with relatively low refractive indexes like quartz and beryl can aggravate that problem.

Gem materials exhibit characteristic refractive indexes which vary from species to species. Light rays are refracted (bent) when they pass into and out of different mediums (transparent materials). Light enters and exits gemstones through refraction. Depending on its incident angle with the surface of a facet and the refractive index of the material, light striking the facet will either reflect back off the surface of the facet, or refract and pass through it. The dividing line between reflection and refraction is known as the critical angle of the material, which varies from one species to another as a simple function of their refractive indexes, n:
critical angle = arcsin(1/n) which is also commonly expressed as sin-1(1/n)

Pavilion facets do their job by reflecting light which has entered the gemstone and is propagating through the gem material. If the geometry of the pavilion facets is such that light strikes them at an angle below the critical angle, it is refracted and leaks out of the pavilion, rather than being reflected and returned to the viewer through the crown of the gemstone. A very simple gemstone model can be employed with a ray tracer as a model to illustrate how light is leaked from the pavilion when its facet angles are too shallow or too steep. This model and ray traces were generated using Robert Strickland's GemCad Software and its 'Waytwace' function for a refractive index of 1.54 (quartz) which has a corresponding critical angle of about 40.49 degrees.

While most real world gemstones and light sources are more complex than this model, reducing the pavilion to a single course of facets and simplifying the "crown" to a 100% table for modeling purposes helps visualize the fundamental effect of varying pavilion angles and serves as a good first approximation for the real world results.

When incident light propagating through quartz strikes a pavilion facet at an angle below 40.49 degrees, it refracts and exits the stone through the pavilion, rather than being reflected and returned to the crown. Incident light striking a facet at an angle equal to the quartz's critical angle travels along the surface of the facet and is lost at the culet in this model. When incident light propagating through quartz strikes pavilion facets at an angle above 40.49 degrees, it is internally reflected and strikes another facet. Reflection is a symmetrical phenomena, that is, the incident and reflected angles are equal. Refraction is usually asymmetrical, the incident and refracted angles being equal only in special cases, such as when light waves are traveling parallel (rays perpendicular) to the refracting surface.

The result when internally reflected light strikes the next facet in its path depends on its angle of incidence relative to that facet. If that angle is greater than the critical angle, the light is reflected once again. But if that angle is below the critical angle, the light gets refracted by the next facet and exits the stone through it. The geometry of the model is such that for a refractive index of 1.54 (quartz) and pavilion angles above the critical angle but below about 47 degrees, the incident light is reflected back and returned to the table.

For angles exceeding 47 degrees, incident light is reflected when it strikes the first pavilion facet. However with pavilion angles above 47 degrees the geometry changes such that the reflected light strikes the second pavilion facet with an angle of incidence below the critical angle, becoming refracted and lost. I will spare the reader hard numbers here, but if you're interested in them, a more detailed treatment and mathematical description of the physics of refraction and calculating raypaths, based primarily on Snell's Law, is online in Refractive Index and Critical Angle. If your math is a little weak or rusty, don't despair as the general principles at play can be grasped and appreciated without employing trigonometry and working specific examples through the underlying formulas. It's worth noting that with our simple model, quartz exhibits an envelope of pavilion angles that's less than 7 degrees wide (40.5 - 47 degrees) where incident light is reflected via both sides of the pavilion and returned to the crown.

The higher a material's refractive index, the lower its critical angle. One result of that relationship is the higher the material's refractive index, the wider the envelope of pavilion angles that will produce reflection and return light to the crown of the stone, rather than leaking out due to the pavilion angles being too shallow or too steep. A similar but significantly wider envelope of angles for the round model shown above extends from about 27 to 51 degrees for a material with a refractive index in the neighborhood of 2.20, close to Cubic Zirconia's. A general consequence for design is that lower RI materials such as quartz or beryl exhibit narrower and more restricted envelopes for pavilion angles that will return light to the crown.

Novelty and special effect type cuts excepted, the generally desirable effect is of course to have the entire stone appear 'bright' and 'lively', and without dark areas of major extent. In terms of desirable optical characteristics, the brilliant cut round is a pretty tough generic design and shape to beat. Due to the high symmetry, the pavilion facets on a round can all be placed near optimal angles for reflecting and designed so that a relatively high percentage of light entering the gemstone is returned to the viewer. Stones that behave this way optically are perceived to be bright, one reason brilliant cut rounds are so popular and prevalently employed in jewelry.

RandomCosine = 87.9ISO = 93.4
p41.00°3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93
g90.00°3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93
b36.00°3,9,15,21,27,33,39,45,51,57,63,69,75,81,87,93
m32.00°12,24,36,48,60,72,84,96
s18.00°6,18,30,42,54,66,78,90
t00.00°Table
t00.00°Table

A simple round design produces these random, Cosine and ISO brightness plots for quartz (RI = 1.54), with GemRay calculating a Cosine light model index of 87.9 and an ISO light model index of 93.4 using the angles indicated. This is a very bright stone. No bow ties here!

Tourmaline image courtesy www.faceters.com.
Gemstone design would be much simpler if all rough was round and brightness was the sole figure of merit for gemstones. Enhancement and display of color, dispersion, scintillation, tilt performance, and of course shape of the rough combined with a desire for high yield with precious materials all come into play. In the matter of brightness, gemstone design is somewhat a matter of wanting to have your cake and eat it too, in that other desiderata are often paid for with brightness. However, gemstones suffer if they are too dark overall due to too much brightness being traded off, or just plain leaked out. Brightness is what you've got left to have fun with after paying the bills.

Unfortunately the high symmetry facilitating the optical performance of brilliant cut rounds can be less than visually interesting. While their optical performance can be very high in the photon return department, there's not much unique about the shape of a round brilliant. There is also considerable motivation from the perspective of yield to cut other than round shapes due to typical shapes of crystals for various gem species. For instance, tourmaline crystals are often naturally shaped such that a stone with a long rectangular shape produces the least waste and highest yield. Maximum size and yield are major considerations in matching a shape and design to a particular piece of gem rough, particularly so with pricier gem materials.

RandomCosine = 87.8ISO = 92.1
p41.00°24,48,72,96
g90.00°24,48,72,96
c140.00°24,48,72,96
c230.00°24,48,72,96
c320.00°24,48,72,96
t00.00°Table

Employing 41° pavilion facets, with 40°, 30° and 20°, crown steps for a simple square design produces these random, Cosine and ISO brightness plots for quartz (RI = 1.54), with GemRay calculating a Cosine light model index of 87.8 and an ISO light model index of 92.1.

Rounds and squares are fine you say, but what about those rectangular shaped points of Mount Ida's finest in your rough box? Could you simply "stretch" a design like the simple square to fit them?

RandomCosine = 61.6ISO = 70.1
p152.51°48,96
p241.00°24,72
g1-g290.00°48,96-24,72
c1-c240.00°48,96-24,72
c3-c430.00°48,96-24,72
c5-c620.00°48,96-24,72
t00.00°Table

Oops, check out those dark regions - we've just developed a case of the bow tie blues! Note that stretching out the shape reduces the symmetry and requires cutting pairs of pavilion facets at different angles to meet the girdle and culet point. The angles of the facet pair spanning the narrow dimension of the rectangle must be steeper than the angles of the facet pair spanning the wide dimension. If the shallowest facets are cut so that incident light strikes them at less than the critical angle they will refract, or "window" and leak the light out of the pavilion.

If the shallower pair of pavilion facets for a simple rectangle with L/W = 1.5 are cut at 41° to keep them just above the critical angle for quartz, the steeper pair must be cut at 52.51° to meet with them at the culet. If you'll recall, our simple gemstone model exhibited a working pavilion angle envelope of less than 7° and that it began leaking light when the pavilion angles exceeded 47° with quartz.

There's more than 11° difference between the 41° and the 52.51° facets on this 1.5:1 rectangle, and the 52.51° facets are more than 5° steeper than the upper limit of the envelope for the angle where light began leaking in the quartz model. It doesn't take a rock scientist to see where those bow tie shaped dark areas are coming from.

This fundamental geometrical problem is of course not limited to rectangles and is manifest in other common gemstone shapes such as ovals and the marquise. The higher the length to width ratio of the shape, the greater the angular difference required between pavilion end facets and pavilion side facets connecting a level girdle to a central culet point.

Here's sequence of GemRay brightness plots serving to illustrate the aggravating effect of increasing length to width ratios as the simple round design example presented above is progressively scaled or "stretched" from a round (L/W = 1.0) to an oval with a L/W = 1.5 while holding each oval's shallowest pavilion angle just above the critical angle for quartz at 41.00°.

RandomCosine = 87.9ISO = 93.4 RandomCosine = 71.1ISO = 79.1
RandomCosine = 56.8ISO = 67.2 RandomCosine = 45.2ISO = 58.5
RandomCosine = 38.4ISO = 55.3 RandomCosine = 30.7ISO = 50.1

RandomCosine = 30.8ISO = 47.8
p141.00°24-72
p248.82°08-40-56-88
p352.65°96-48
p454.94°02-46-50-94
p552.36°06-42-54-90
p650.02°11-37-59-85
p745.77°19-29-67-77
a40.00°2-6-11-19-etc.
b30.00°2-6-11-19-etc.
c20.00°2-6-11-19-etc.
t00.00°Table
RI = 1.54

When the shallowest pavilion main (p1) on this simple oval design is set just above the critical angle for quartz at 41.00°, the balance of this design's pavilion geometry dictates a steepest pavilion main (p3) of 52.65°, a spread of pavilion main angles exceeding eleven degrees. While the simple oval's pavilion is a little more gussied up with break facets, the same bow tie generating geometrical and optical problems are at play here.

RandomCosine = 38.1ISO = 54.9
p133.47°24-72
p241.00°08-40-56-88
p344.90°96-48
p447.30°02-46-50-94
p544.60°06-42-54-90
p642.20°11-37-59-85
p738.00°19-29-67-77
t00.00°Table
a Crown Steps40.00°2-6-11-19-etc.
b Crown Steps30.00°2-6-11-19-etc.
c Crown Steps20.00°2-6-11-19-etc.
Table00.00° 

One approach to dealing with the problem of light leaking out of way too steep pavilion facets is to cut the shallowest facets below the critical angle. The general idea here seems to be to sacrifice the light return from the shallowest facets to lower the angles of the balance of the facets and bring the rest of them within a functional performance envelope. In the example above, the pavilion angles of the simple oval are changed so that the p2 facets are cut at 41° producing 33.47° facets for the shallowest mains (p1) and 44.90° for the steepest mains (p3). This sacrifical tactic can improve the overall brightness of a pavilion with bow tie problems, but it is somewhat akin to robbing Peter to pay Paul and practically guarantees a bow tie due to light leaking out of the pavilion facets below the critical angle. When that happens the bow tie is just worn a little differently.

RandomCosine = 40.1ISO = 57.6
p133.47°24-72
p241.00°08-40-56-88
p344.90°96-48
p447.30°02-46-50-94
p544.60°06-42-54-90
p642.20°11-37-59-85
p738.00°19-29-67-77
t00.00°Table
a Crown Steps40.00°2-6-11-19-etc.
b Crown Steps30.00°2-6-11-19-etc.
c Crown Steps20.00°2-6-11-19-etc.
Table00.00° 

The higher the refractive index materials can be more forgiving from a design point of view regarding bow tie problems, in that higher RIs provide wider envelopes of functional pavilion angles. Light tends be scattered about more as it reflects around in gemstones cut from higher RI materials due to the lower critical angles. Lower critical angles equate to wider cones of "acceptance" for light striking a facet externally, and wider cones of reflection for light once it has refracted into the gemstone and is propagating and bouncing around internally. The distribution of brightness is also an important factor and consideration affecting the overall appearance and perception of a gemstone. The example above employs the same angles as the preceding variation of the simple oval, but was calculated by GemRay for a material with an RI of 2.20 instead of 1.54.

While not much brighter overall in the preceding example where facets are "windowing" due to being cut below the critical angle for quartz, the distribution of brightness in the raytraces of the simple oval design is significantly more uniform for the higher RI material. The dark areas are smaller in extent, more numerous and less concentrated than with quartz. As previously noted, other desiderata such as yield, presentation of color and tiltability are paid for with brightness. A gemstone design doesn't have to be extremely bright to be a "good" design, just like an car doesn't have to be extremely fast to be a "good" car. It all depends on where the faceter is trying to go and wants to get done along the way. Some novel patterns and cuts make use of intentionally designed leaks to produce special effects like a bright star "floating" in a dark background, but black holes due to bow tie prone geometries are generally undesirable.

Because the distribution of brightness changes with refractive index, a design that performs adequately in high RI materials such as CZ (Cubic Zirconia - a common diamond substitute and popular synthetic material in it's own right with hobbyist faceters) can be problematic when a faceter adapts and cuts it in low RI materials such as quartz or beryl. A pitfall and gotcha that has no doubt resulted in many bow tie afflicted quartzes, beryls and other low RI gemstones over the years is cutting designs for ovals, rectangles, marquise, and other shapes of increasing L/W that were originally cut in and recommended on the basis of results with a high RI material.

If the design has a bow tie prone geometry, the faceter is damned going in if he uses pavilion angles below the critical angle of the material he's cutting. However, if he uses tangent ratio to convert the design angles so they all operate above the critical angle of his lower RI material, the spread of mains angles often extends significantly beyond the rather narrow performance envelope of workable angles for a low RI material. The black holes created by light leaking out the stones are just as black no matter how many times it gets reflected internally when it leaks out the pavilion instead of being returned to the viewer through the crown.

There are "brighter" tactics and general design solutions for dealing with this geometrical problem. While the underlying physics and math are what they are, there are several possible approaches to making favorable changes to the problematic geometry by abandoning paradigms such as the level girdle and the point cutlet.

RandomCosine = 83.3ISO = 88.3
p41.00°24,48,72,96
g1-g290.00°48,96-24,72
c1-c240.00°48,96-24,72
c3-c430.00°48,96-24,72
c5-c620.00°48,96-24,72
t00.00°Table

If one is willing to forego a conventional girdle of uniform thickness all around and allow the girdle facets running the length of the stone to get thick, the length-to-width ratio of the shape ceases to dictate the geometry and those troublesome too-steep facets spanning the width of the stone can simply be replaced with facets cut at an optimal angle. Unfortunately, such thick girdle segments aren't very practical as they would cause significant mounting difficulties for many types of settings. They also create a rather clunky and less than elegant overall appearance that's unappealing to many tastes even when mounting ramifications are not at play.

However it is not necessary to forego the paradigm of a conventional girdle to fix pavilion geometries underlying the bow tie effect. In Part II we'll take a look at how keels can replace pointed culets and cone shaped pavilions to provide some geometrical relief as the length to width ratios of gemstone shapes increase. We'll also explore Barion style designs as another approach and solution addressing the bow tie effect while retaining pointed culets and other optical advantages of cone shaped pavilions as the length to width ratios of gemstone shapes increase.

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