Total Internal Reflection & Critical Angle | Physics Tutorial

Gemstone cutting relies on the physics of light refraction and reflection, where precise angles at the pavilion (bottom) of a stone determine its brilliance by catching and reflecting light back to the viewer; cutters must balance maximizing rough stone weight with creating optimal design, and can use techniques like the American brilliant pattern (originating from the glass industry) to enhance color and visual appeal through strategic facet placement.

Gemstones must be cut at precise angles to display their brilliance. When cut correctly, light enters the stone and reflects back to the viewer's eye. A perfectly cut diamond displays all colors of the rainbow because the angles are cut at the correct angles to maximize light refraction and dispersion. This demonstrates how geometric precision in stone cutting directly affects visual properties.

Gemstone cutting is a precise science combining physics, mathematics, and artistry. The critical angle (372 degrees for sapphire) determines how light enters and exits the stone, with the bottom third (pavilion) creating all brilliance through light reflection. When the culet is off-center, the stone becomes 'windowed' and loses life. Cutters use step cutting (5-degree increments) to adjust angles while minimizing weight loss. The refractive index (1.76 for sapphire) slows light, causing it to bounce back through the table to the viewer's eye. For negative carving, inverted V-shapes are cut at precise critical angles to create brilliance. The pavilion angle at the culet is typically 36.5-37 degrees, calculated from the critical angle of 34 degrees. This allows cutters to 'squeeze' the stone, preserving weight while maintaining optical performance.

The brilliance and sparkle of diamonds is due to total internal reflection. When light enters a diamond, it undergoes multiple total internal reflections before exiting, which creates the characteristic sparkle. This is why diamonds are cut with specific angles to maximize this effect.

The brilliance of a diamond is due to total internal reflection. Diamond has a very high refractive index (approximately 2.42), resulting in a very small critical angle of about 24°. The diamond is cut with specific angles so that light entering the diamond strikes the internal surfaces at angles greater than 24°, causing total internal reflection. This traps most of the light inside the diamond, and only a small fraction escapes, creating the sparkling effect. The sharp edges and precise cutting angles are essential for this effect.

Mirage occurs due to refraction and total internal reflection in air layers of different temperatures. Near hot surfaces, air becomes less dense (lower refractive index) while cooler air above is denser. Light from the sky bends away from the normal, creating illusions of water on hot surfaces. Periscopes use prisms with total internal reflection for observation from hidden positions, commonly used in submarines. Rainbows form when white light enters water droplets and undergoes dispersion followed by total internal reflection. Primary rainbow involves one internal reflection at 42 degrees, while secondary rainbow involves two reflections at 51 degrees with reversed colors.

Mirages are optical illusions caused by the refraction of light through air layers of different temperatures and densities. Light travels in straight lines only in homogeneous media but bends when passing between layers of different densities. The most common type is the lower mirage (road mirage), where hot ground heats the air near the surface, causing light from distant objects to bend downward and create the illusion of water reflecting the sky. Upper mirages occur when cold air near the surface is overlaid by warmer air above, revealing objects beyond the horizon and explaining legends of flying ships. Lateral mirages form when air layers are positioned at unusual angles, creating distorted reflections. These phenomena demonstrate how temperature gradients in the atmosphere can dramatically alter the apparent position and appearance of distant objects.

Mirages form when light passes through air layers of different temperatures (and thus different refractive indices). Rainbows form when sunlight enters raindrops: refraction, dispersion (splitting into colors), total internal reflection, and final refraction. Both phenomena demonstrate how light bends through media with varying optical properties.

When light travels through air of different temperatures, it can curve upward and reflect rather than hitting the surface, creating mirages. This is particularly common over large bodies of water where heat rising off the water can cause light to bend. A famous example is the boating scene from the Pierce Brosnan remake of The Thomas Crown Affair, where the New York City skyline was visible due to perfect mirage conditions despite being too far to see normally.

Rainbows form through a combination of dispersion, refraction, and total internal reflection in water droplets. Sunlight enters a droplet, undergoes refraction and dispersion (splitting into colors), then total internal reflection inside the droplet, and finally refraction again as it exits. This produces the colorful arc visible after rainfall. Mirages occur when light bends through air layers of different temperatures. In hot conditions, air near the ground is less dense than air above, causing light from distant objects to bend upward and creating illusions of water on roads or objects appearing below their actual position. These phenomena demonstrate how atmospheric conditions affect light propagation.

Reflecting prisms use total internal reflection to redirect light: (1) Porro prisms in binoculars reflect light 180° to fold the optical path, (2) Right-angle prisms reflect light 90° for periscopes, (3) Porro prisms in binoculars provide erect images. Prisms are preferred over mirrors because they reflect 100% of light without coating degradation.

Reflecting prisms use total internal reflection to redirect light. A right-angled prism can reflect light by 90° or 180°. The advantage over mirrors is that prisms do not suffer from absorption losses and maintain image quality. They are used in binoculars, periscopes, and other optical instruments.

A total internal reflecting prism is an isosceles right triangle (90-45-45 degrees) used to reflect light through total internal reflection. When light enters perpendicular to one leg, it passes through without refraction. When it strikes the hypotenuse at 45 degrees, total internal reflection occurs because 45° exceeds the critical angle of 42° for glass-air interface. Light exits perpendicular to the other leg, emerging parallel to its original direction. This device redirects light paths in periscopes, binoculars, cameras, and other optical instruments. A periscope uses two such prisms: the top prism reflects light downward, and the bottom prism reflects light horizontally toward the observer's eye, allowing observation from a lower position.

A right-angled isosceles prism (90°, 45°, 45°) changes light direction by 90° when light enters perpendicular to one leg and strikes the hypotenuse at 45°. Since the critical angle for glass is ~42°, total internal reflection occurs. This principle is used in periscopes and binoculars. Two prisms in binoculars redirect light to create a compact design while maintaining the same magnification as a longer telescope. Reflecting prisms are preferred over mirrors because they are more durable, maintain consistent reflectivity over time, provide a wider field of view, and can be coated with reflective materials like aluminum oxide or magnesium fluoride.

Reflecting prisms are right-angled glass prisms (n=1.5) used to redirect light by 90° or 180° through total internal reflection. The critical angle for glass-air is approximately 41.8°. When light enters perpendicular to one face, it strikes the hypotenuse at 45°, exceeding the critical angle, causing total internal reflection. Prisms are preferred over mirrors because they cause less light loss (no metallic coating degradation). The instructor explains how prisms work in periscopes, binoculars, and other optical instruments. The section also covers coating prisms with thin transparent layers (fluoride) to further reduce light loss during entry and exit.

Optical tunnelling, or frustrated total internal reflection (FTIR), occurs when an evanescent wave that normally decays to zero at the boundary between two media can penetrate into a third medium if the distance between the first two media is sufficiently small; this phenomenon allows light to 'tunnel' through what would otherwise be an opaque barrier, enabling applications such as beam splitters where incident light is divided into reflected and transmitted components based on the gap width and refractive indices of the materials involved.

Frustrated total internal reflection occurs when an evanescent wave, which is a standing electromagnetic wave that forms at the boundary during total internal reflection, can transfer energy to a nearby medium if that medium is brought within a distance less than the wavelength of the radiation; this principle was demonstrated using paraffin wax prisms with microwaves, where bringing a second prism close enough allowed microwaves to pass through rather than reflecting, unlike with glass prisms where much closer proximity (sometimes requiring water) was needed due to the much shorter wavelength of visible light compared to microwaves.

This section explores advanced wave phenomena arising from electromagnetic boundary conditions. It covers: (1) The mathematical conditions for total internal reflection when sin(θ_I) > n_T/n_I; (2) The transformation of transmitted waves into evanescent waves with exponential decay in the direction perpendicular to the interface; (3) The calculation of decay constants and propagation constants for evanescent waves; (4) The critical insight that while evanescent waves exist, no net energy flows across the interface, demonstrating that total internal reflection involves complete reflection with no power transmission.

Frustrated total internal reflection is a wave phenomenon where waves that would normally undergo complete reflection at an interface can instead transmit through a small gap between two media when brought sufficiently close together; this occurs because the evanescent wave that exists just beyond the critical angle can couple across the gap, allowing energy to pass through even though the incident angle is greater than the critical angle for simple total internal reflection.

During total internal reflection, an evanescent wave exists in the lower-index medium with exponentially decaying amplitude (e^(-|kz|z)). This wave does not transfer energy across the interface because the Poynting vector has no normal component. Frustrated total internal reflection occurs when a thin layer separates the two media, allowing the evanescent wave to tunnel through and couple to the third medium. This principle enables fingerprint detection: ridges make contact and frustrate total internal reflection, while valleys maintain it, creating high optical contrast.