Comprehensive Study Notes on Total Internal Reflection and Its Applications

Sparkle of Cut Diamonds

A diamond is well-known for its brilliant sparkle, which is a direct consequence of the principle of total internal reflection. This optical phenomenon is facilitated by the diamond's exceptionally high refractive index, which is approximately $2.42$. Because of this high refractive index, the critical angle for a light ray traveling from the diamond medium into the air medium is very low, measuring only about $24^{\circ}$. Diamonds are meticulously cut into specific shapes with various faces to maximize this effect.

When a ray of light enters the cut diamond, it strikes the internal faces. Due to the very low critical angle of $24^{\circ}$, the light often hits these faces at angles of incidence that are greater than the critical angle. This causes the light to undergo a large number of total internal reflections at various faces before it can eventually emerge. This process effectively traps the light within the crystal for some time, causing the diamond to sparkle. To the eyes of an observer, the diamond appears bright when the angle of incidence at any specific face is less than $24^{\circ}$, because at that point, the light emerges from the facet in its entirety.

Mirror-Like Appearance of Water Surfaces

Under certain conditions, the upper surface of water, such as that contained in a beaker, can appear like a mirror. This effect is observed when the beaker is held above eye level and viewed from below. Light rays that emerge from below the surface of the water undergo refraction as they move toward the air. However, if the rays are incident at an angle greater than $48^{\circ}$, total internal reflection occurs because the critical angle for water is exactly $48^{\circ}$. Consequently, the light rays are reflected back into the water and appear to come from the upper surface. This causes the surface to take on a silvery, mirror-like appearance.

Mirage Formation in Deserts

A mirage is an optical illusion frequently seen in desert environments that creates the false perception of the presence of water. This phenomenon is a result of temperature and density gradients in the air. In a desert, the sand becomes significantly heated by sunlight, which in turn heats the lower layers of the air. As the height above the sand increases, the layers of air become progressively cooler. This temperature gradient results in a density gradient: the hot air near the ground is a rarer medium (less dense), while the cooler air above it is a denser medium.

When a light ray proceeds downwards from a distant object, such as a tree, it passes through layers of air with gradually decreasing density. At every interface between these layers, refraction takes place, and the light ray bends away from the normal as it moves from a denser to a rarer medium. The angle of incidence continues to increase for the ray until it reaches a specific layer where the angle of incidence exceeds the critical angle for that air layer interface. At this point, the light ray undergoes total internal reflection and begins to travel upward.

As the ray moves upward from rarer to denser layers of air, it bends toward the normal. This makes the entire path of the ray concave in shape. To an observer, these rays appear to originate from a source that is a laterally inverted virtual image of the original object. The observer "sees" this image as if it were reflected from a surface of water, even though no such water exists in the desert.

Silvery Appearance of Cracks and Air Bubbles

The principle of total internal reflection also explains why a crack in a glass window pane appears silvery. Air becomes trapped within the cracks of the glass. When light hits these air-filled gaps, it is bent to such an extent that total internal reflection occurs, resulting in a silvery look. This same explanation applies to the sparkle observed in various cut glass articles.

Similarly, an air bubble rising through a water tank can appear silvery when viewed from above. As light passes from the water (the denser medium) into the air bubble (the rarer medium), it may strike the interface at an angle of incidence greater than the critical angle for water, which is $48^{\circ}$. When this occurs, the light undergoes total internal reflection at the bubble's surface, causing it to appear bright and silvery.

Optical Fiber Technology

Optical fiber is a modern technology used extensively in telecommunications, including internet, broadband, phone lines, and networking. These fibers provide higher bandwidth and can transmit data at much higher speeds than traditional copper cables. The functionality of optical fiber is based entirely on the principle of total internal reflection.

An optical fiber consists of a central core and an outer cladding layer. These two parts are manufactured with specific refractive indices to ensure total internal reflection occurs. In a typical configuration, the core is made of quartz glass with a refractive index of $1.7$. This core is coated or cladded with a material that has a lower refractive index of $1.5$. When a light signal enters the fiber within a specific angular range, it strikes the core-cladding boundary at an angle greater than the critical angle. Consequently, the light undergoes repeated total internal reflections as it travels through the core.

This method allows light signals to be transferred over long distances with negligible loss of energy. Aside from telecommunications, optical fibers are also utilized in medical diagnostics for checking the internal parts of the human body through a procedure known as an endoscopy test.

Vision of a Fish in Water

Total internal reflection significantly impacts how aquatic life, such as a fish, perceives the environment above the water. A fish can see the entire $180^{\circ}$ view of everything above the water surface within a condensed cone of only $96^{\circ}$. This is because the critical angle for the water-air surface is $48^{\circ}$. The field of vision for the fish is twice this critical angle, resulting in a visual cone of $2 \times 48^{\circ} = 96^{\circ}$. Outside of this $96^{\circ}$ cone, light from above cannot reach the fish's eyes via refraction, and the surface may instead reflect light from within the water back down to the fish.