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Refractive Index and Critical Angle
by Bob Keller
You will often encounter the terms refractive index and critical angle employed in faceting related discussions. Understanding these terms and how they relate to the optical properties of gemstones and different gem materials is fundamental to faceting, designing gemstones, and adapting designs optimized for one material to another.
The speed of light is not constant - it varies as it passes through different transparent substances. Light travels more slowly through air than it does through the vacuum of empty space, more slowly through water than air, slower yet through quartz, and slower yet through diamond. The speed of light is equal to its wavelength times its frequency. As light passes through different substances or mediums its frequency remains constant and its wavelength changes. This change of wavelength (speed) at the surface interface between different mediums causes light passing through one and into the other to be bent - to be refracted.
You are already familiar with common manifestations of this phenomena. 'White' light is composed of light containing waves at all the visible frequencies. When white light is refracted, as when passing through a prism, waves of different frequencies are bent at different angles, with the result being that the light is dispersed - spread out into its component colors. When the refracting interface is formed by droplets of water in the air, the result is a rainbow.
Another commonly experienced effect of refraction is the apparent foreshortening of distance, or 'magnifying' effect that occurs when an underwater object is viewed from above the water's surface. When the refracting interface is the surface of a lake, the observed object is a fish which unhooked itself before being pulled through the surface, and the observer is a fisherman, the result is a whopper.
The index of refraction or refractive index (R.I.) of a particular substance is equal to c (the speed of light in empty space) divided by the speed of light in that particular substance. The speed of light is greatest in empty space - approximately 300,000,000 meters per second. Since the speed of light is reduced when it propagates through transparent gasses, liquids and solids, the refractive index of these substances is always greater than 1. The list below gives the refractive indices of several familiar substances.
Substance State Refractive Index Air Gas 1.000293 Ice Solid 1.31 Water Liquid 1.33 Ethyl Alcohol Liquid 1.36 Fluorite Solid 1.43 Quartz Solid 1.54 Salt Solid 1.54 Tourmaline Solid 1.62 Garnet Solid 1.73-1.89 Cubic Zirconia Solid 2.14 - 2.20 Diamond Solid 2.41
The higher the R.I. of a substance, the greater the angle that light bends at the surface interface between air and the substance. Light travels nearly as fast in air as it does in empty space and is not much bent by air. For purposes regarding practical calculations and considerations involving refractive indices, faceters can treat air as empty space.
However, light travels 1.54 times faster through empty space or air than it does in quartz and 2.41 times faster through air than diamond. Light is bent or refracted to a greater degree by diamond than quartz, and at various intermediate angles by other gemstone materials such as beryl (R.I. = 1.58), tourmaline (R.I = 1.62) and garnet (R.I. = 1.73-1.89).
The following construction illustrates the phenomena of refraction with a graphical representation of wavefronts of light traveling through space and the resulting refraction as they pass from space into a transparent material (blue area) - such as water, or aquamarine. The construction is used to derive Snell's Law, a fundamental law of geometrical optics.
The math in the following examples is provided for those who appreciate it, but don't despair if your algebra and trig are a little rusty. A common misconception about faceting is that it requires a lot of math. You can be an excellent faceter with very limited math skills. The important thing is to understand the underlying principles and effects at the concept level.
The difference in refractive indices between materials has significant bearing on the design of gemstones, selecting material appropriate to a particular design, and converting design angles to work properly with different gem materials.
Light is not only bent or refracted when it enters a gemstone, but also when it has traversed the stone and strikes the internal side of the surface interface between the gem material and the surrounding air. Depending on the R.I. of the material and the angle at which the wavefront strikes the internal surface of a facet, the wavefront will either be refracted and exit the gemstone by passing through the facet back into the air, or it will be reflected back into the gemstone.
Gemstones are generally designed so that light entering them from the viewing side (the crown) is internally reflected by facets on its back side (the pavilion) and directed to exit the gem through facets on the crown to the eye of the viewer. If a significant amount of the light entering the gem from the crown 'leaks' out by being refracted rather than reflected by the pavilion facets, the brightness of the gem suffers. The angle at which light is internally reflected rather than refracted is known as the critical angle. This angle varies for different materials and is a function of their refractive indices.
The percentage of light that is reflected by the pavilion facets and returned through the crown facets is considered to be a figure of merit for various designs. Generally, the less light lost out the pavilion the better. If pavilion facets of a stone are cut below the critical angle for the material, light will pass through them like a window rather than being returned to the viewer through the crown. Hence the term windowed stone. The pavilion of a gemstone is designed to act like a mirror for light entering the crown of the stone.
The formula equating the critical angle to n (refractive index) is critical angle = sin-1(1 / n). Wavefronts of light striking interior surface of a facet at an angle less than the critical angle are refracted through the surface interface and exit the stone. Wavefronts of light striking the interior surface of a facet at greater than the critical angle will be reflected back into the gemstone.
Reflection is symmetrical - wavefronts which are reflected from a surface form the same angle to it coming and going regardless of the refractive index of the material.
Refraction is not symmetrical, when a wavefront passes through the surface of a facet it comes and goes at different angles to the surface. The difference in angles formed by a wavefront with a refracting surface is a function of the refractive indexes of both mediums.
A special case is a wavefront propagating parallel (light ray perpendicular) to the surface of a facet. In this case the angle of incidence between the wavefront and the surface is zero, so the angle of refraction is also zero.
Light rays are constructs used in geometrical optics to aid in the visualization and analysis of optical systems by ray tracing. Light rays are perpendicular to wavefronts. So the angle of incidence formed by a light ray to the normal of a refracting/reflecting surface is equal to the angle of incidence formed by its wavefront with the surface. A normal is math nomenclature for a line perpendicular to another one. Light rays form the same angle with the normal to a surface as a wavefront forms with the surface, because light rays are perpendicular to wavefronts.
The same laws and formulas that apply to wavefronts apply to light rays - but because light rays are perpendicular to wavefronts, the angle formed between the ray and the normal to the surface is substituted for the angle formed between the wavefront and the surface. The angle between the ray and the normal is equal to 90° minus the angle between the ray and the surface. The equality of the angles between a ray and the normal to a surface and a wavefront and the surface is shown geometrically in the following illustration.
Let's examine several examples to see the effect of refractive index and critical angle. In the illustration below, a light ray has entered through the crown of a stone and is traveling through it on a path that's parallel to the axis of the stone. When the ray strikes the internal side of the pavilion, it will be refracted and lost if the angle of incidence it forms with the normal to the the facet is less than the critical angle. For quartz, the critical angle is sin-1(1 / 1.54) = sin-1(.649351) = 40.49°.
In this case the pavilion facets are cut at 35°. The light ray forms a 55° angle with the pavilion facet, so the angle it forms with the normal to the facet is (90 - 55)° = 35°. 35° is below the critical angle for quartz so the ray is refracted and lost through the pavilion facet. :( Snell's Law is used to solve for the angle of refraction, which is 62.04° in this case. The exit angle x formed between the ray and the external surface of the pavilion facet is equal to 27.96°
In the next example the pavilion facets are cut at 43°. A light ray traveling through it on a path that's parallel to the axis of the stone strikes the pavilion facet at a 47° angle. The angle it forms with the normal to the facet is (90 - 47)° = 43°. 43° is above the critical angle for quartz, so the ray is reflected by the first pavilion facet it strikes. Rays reflect symmetrically regardless of the refractive index of the material, so the ray is reflected by the first pavilion facet at a 47° angle. It then travels through the stone to a pavilion facet on the opposite side.
Since the pavilion facets are cut at 43° the culet angle between them is 94° because (180 - 2(43))° = 94°. The angle of reflection formed by the ray and the first facet it strikes is 47°. Note that the pavilion faces and the lowermost segment of the light ray form a triangle. Since the sum of the interior angles of any triangle = 180°, the angle the ray makes with the second pavilion facet is (180 - 47 - 94)° = 39°.
The angle between the ray and the normal of the second facet = (90 -39)° = 51°. 51° is above the critical angle for quartz, so the ray is again reflected and returned to the crown. :)
In the next example the pavilion facets are cut at 50°. A light ray traveling through it on a path that's parallel to the axis of the stone strikes the pavilion facet at a 40° angle. The angle if forms with the normal to the facet is (90 - 40)° = 50°. 50° is above the critical angle for quartz, so the ray is reflected by the first pavilion facet it strikes at a 40° angle. It then travels through the stone to a pavilion facet on the opposite side.
Since the pavilion facets are cut at 50° the culet angle between them is 80° because (180 - 2(50))° = 80° The angle of reflection formed by the ray and the first facet it strikes is 40°. Note that the pavilion faces and the lowermost segment of the light ray form a triangle. Since the sum of the interior angles of any triangle = 180°, the angle the ray makes with the internal surface second pavilion facet is (180 - 40 - 80)° = 60°.
The light ray forms a 60° angle with the second pavilion facet, so the angle it forms with the normal to the facet is (90 - 60)° = 30°. 30° is below the critical angle for quartz, so the ray is refracted and lost through the second pavilion facet. :( Snell's Law is used to solve for the angle of refraction, which is 50.35° in this case. The exit angle x formed between the ray and the external surface of the pavilion facet is equal to 39.65°
It is instructive to note that if the material in this example is changed to one with a critical angle below thirty degrees, then the ray will be reflected and returned to the crown rather than refracted by the second pavilion facet and lost. The formula equating n (refractive index) to critical angle is n = 1 / sin(critical angle). In this example, gem materials with n = 1 / sin(30°) = 2.00 or greater will reflect the ray from the second pavilion. Cubic Zirconia, a commonly faceted synthetic with an R.I around 2.14-2.20 would return the ray in this example to the crown rather than losing it to refraction as does quartz.
Summary of Some Key Fundamental Concepts
- The speed of light varies in different transparent materials. Each material has a characteristic refractive index, which is the ratio of the speed of light in empty space to the speed of light in the material. Because the speed of light is greatest in empty space, the refractive index of a material is always greater than 1.
- For practical computational purposes, faceters can treat air as empty space.
- Gemstones are generally designed to reflect internal rays from their pavilion facets. The less light lost to refraction through the pavilion facets, the more returned to the viewer through the crown and the brighter the stone.
- The critical angle of a material determines whether an internal ray will be reflected back into a gemstone or refracted by a facet. The critical angle of a material is a function of its refractive index. The higher the refractive index, the lower the critical angle.
- A gemstone will window and leak significant amounts of light from its pavilion facets if they are cut below (at an angle numerically less) than the material's critical angle.
Real World Light Sources
The examples used above are illustrative and serve to explain some key concepts regarding refraction and critical angle applied to faceting, but the underlying models are really too simple to use to fully describe the behavior of light in an actual gemstone. The light in real world viewing environments is not coherent, the incoming rays are not parallel, and they originate from numerous sources. Light entering actual gemstones from the crown side is first refracted through a crown facet, and the angle at which a ray is directed towards a pavilion facet is dependent on the angle of the crown facet and the angle with which the ray enters it in addition to the refractive index of the material. Light also enters actual gemstones through girdle and pavilion facets where these are not shadowed or covered by their mountings.
The refractive index of a material is not constant for light of different frequencies (colors), the result being that different colors of light are refracted at different angles. The higher its frequency the slower light travels through a given material. Refractive index increases as the frequency of the light increases. Violet light, which is at the high frequency end of the visible spectrum, is refracted at a greater angle than red light, which is at the low frequency end of the visible spectrum. The variation in refractive index over the entire optical spectrum, which includes ultraviolet light above and infrared light below the visible spectrum, is only about 2 percent. However, that variation is sufficient for a gemstone to separate or disperse the visible wavelengths through differential bending.
The effect of dispersion or the spreading out of light into it's component colors is often referred to as "fire" in a gemstone. The effect of dispersion can be problematic in optical systems such as microscopes or telescopes, and the optical engineers who design them go to great lengths to minimize it. However, dispersion is a highly desirable characteristic and property in gemstones. You can think of the effect of dispersion as a "light shredder" where a parent ray of "white" light striking the refracting interface is spread out into many child rays, each with a different frequency (color), and each propagating into the stone at a different speed and in a different direction. The greater the dispersion of the material, the greater the difference in their speeds and the wider and further apart the child rays spread before they exit the stone and are returned to the viewer. The wider this spread, or "scattering of the children", the greater the "fire" or prismatic effect.
The coefficient of dispersion is given by the difference in a substance's refractive indexes:
(nf - nc) where
nf = the refractive index for light with a wavelength equal to 486.1 nm (the F Fraunhofer line)
nc = the refractive index for light with a wavelength equal to 656.3 nm (the C Fraunhofer line)
The dispersive power of a material is commonly expressed in optical literature as:
(nf - nc) / (nd - 1) where
nf = the refractive index at 486.1 nm (the F Fraunhofer line)
nc = the refractive index at 656.3 nm (the C Fraunhofer line)
nd = the refractive index at 589.3 nm (the D Fraunhofer line)
The plot thickens quite a bit as soon as absorption, ie materials with color, become involved. Dispersion curves were first compiled for materials that are transparent throughout the visible spectrum where any absorption bands were located well beyond the visible region. Under such conditions there is a smooth decrease of refractive index corresponding to decreasing frequency across the entire visible spectrum. Such materials are said to exhibit "normal" dispersion.
When the material has color it has one or more absorption bands within the visible spectrum, and minima and maxima for the refractive index curve occur at frequencies closely adjacent to the absorption bands, with a minima on the high frequency side and a maxima on the low frequency side.
As absorption band frequencies are approached from either side, Sellmeier's formula describes the relationship of refractive index to frequency:
n2 = 1 + ((K*af2) / (af2 - ao2)) where
n = RI
K = a constant
af = adjacent wavelength
ao = wavelength of maximum absorption
The result being that with materials of color the refractive index sometimes increases for longer wavelengths... Even though the effects of color and absorption bands within the visible spectrum on dispersion are well documented and described mathematically, colored materials are said to exhibit "anomalous" dispersion in optical crystallography lingo.
Transparent substances are optically categorized as isotropic or anisotropic. Gases, liquids, glass and other amorphous substances are optically isotropic. The term isotropic is derived from the Greek for "equal turning". In isotropic substances monochromatic light propagates in all directions with equal velocity, the result being that isotropic materials exhibit the same refractive index regardless of the direction light propagates through them. Isotropic materials are termed to be "singly refractive" and they exhibit no birefringence.
Only amorphous gem materials such as opal, and those crystallizing in the cubic, aka isometric crystal system are also optically isotropic. The isometric crystal system is unique in that it embraces all crystalline forms that are referred to three crystallographic axes of equal length and at right angles to each other, with the result that all three crystallographic axes of isometric (cubic) crystals are optically interchangeable.
Diamond, garnet, fluorite, spinel and CZ are gemstone materials belonging to the isometric (cubic) crystal system.
Gem materials crystallizing in the five other crystal systems (tetragonal, hexagonal, orthorhombic, monoclinic and triclinic) are optically anisotropic. The term anisotropic is derived from the Greek for (you guessed it) "unequal turning". In general, the angle with which light is refracted by anisotropic crystals varies some depending on the orientation of the light rays to crystallographic axes of the crystal. Another way of putting this is the speed with which monochromatic light propagates through those materials varies with crystallographic direction. Anisotropic crystals are termed to be "doubly" or "multiply refractive".
Anisotropic crystals are subdivided into the isodametric class, embracing crystals in the tetragonal and hexagonal crystal systems; and the anisometric class, embracing crystals in the orthorhombic, monoclinic and triclinic crystal systems.
Crystals in the isodametric class are referred to two or three equal axes and a third or forth unequal axis at right angles to their plane. The velocity of light in isodametric class crystals is constant in all directions perpendicular (or equally inclined) to a fixed principal axis of crystallographic symmetry. In the direction of this axis, aka the "optic axis", there is no double refraction. Crystals in the isodametric class are said to be "uniaxial".
Crystals in the anisometric class are referred to three unequal crystallographic axes, which may be perpendicular or oblique to one another, depending on whether the material is a member of the orthorhombic, monoclinic or triclinic crystal systems. These crystals have more complex optical relationships than isometric or isodametric anisotropic crystals, but in general they exhibit two directions analogous in character to the single "optic axis" of isodametric class crystals. Hence, crystals in the anisometric class are said to be "biaxial".
The term "birefringence" refers to the difference between the multiple refractive indices exhibited by an optically anisotropic material. Quartz, a member of the hexagonal crystal system and the isodametric class of anisotropic crystals, presents a refractive index nε of 1.55328 to light rays propagating perpendicular (ie: wavefront vibrations parallel) to the crystallographic c-axis (optic axis), and a refractive index nω of 1.54418 to light rays propagating parallel (ie: wavefront vibrations perpendicular) to the c-axis of the crystal. A refractive index intermediate between those values is presented to light rays propagating through quartz in directions between parallel to the c-axis and perpendicular to it.
The birefringence of quartz = nε - nω = 1.55328 - 1.54418 = .0091, a relatively low value - making quartz weakly birefringent.
Beryl, corundum, sphene, kunzite, topaz, tanzanite, peridot, apatite, iolite, zircon and tourmaline are all examples of anisotropic gem materials, and all are birefringent in varying degrees.
The doubly refractive effect in calcite (birefringence = .172) is very strong, as evidenced in the doubled image of print viewed through a crystal of transparent calcite. Materials that exhibit strong birefringence tend to produce fuzzy, out of focus looking gems. Proper orientation of the "optic axis" in gemstones cut from moderate to strongly birefringent materials will help minimize the undesirable "facet doubling" effect of birefringence.
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