Reflection & Refraction

What is Reflection?

Reflection is the bouncing of light off a surface. When light strikes a smooth surface like a mirror, the light bounces back according to the law of reflection. This phenomenon is so familiar—we use mirrors daily—that we often take it for granted. Yet understanding reflection requires grasping the geometry of light rays and the properties of surfaces. Smooth surfaces produce specular reflection, where parallel light rays remain parallel after reflection, enabling clear image formation. Rough surfaces produce diffuse reflection, where light scatters in all directions because the surface normal varies at different points.

The law of reflection states simply: the angle of incidence equals the angle of reflection. Both angles are measured from the normal, a line perpendicular to the surface at the point where light strikes. An incident ray approaching a mirror at 30 degrees to the normal will bounce off at 30 degrees on the other side. The incident ray, reflected ray, and normal all lie in the same plane. This relationship holds regardless of the material or wavelength of light—red and blue light follow the same geometric law.

Reflection is not absorption and re-emission. For metals, electrons oscillate in the electromagnetic field of the light wave, and these oscillating charges radiate light back out. For transparent materials like glass, some light reflects at the surface even though most transmits through. Fresnel's equations describe how much light reflects versus transmits based on angle and polarization. At normal incidence on glass-air interface, roughly 4% of light reflects; at grazing incidence, nearly 100% reflects.

Curved mirrors have unique properties. Concave mirrors (curving inward) focus light to a focal point, enabling applications from shaving mirrors to astronomical telescopes. Convex mirrors (curving outward) diverge light, used in security mirrors and car side mirrors because they show a wide field of view. The focal length depends on the radius of curvature: f = R/2, where R is the mirror's radius. This simple relationship allows precise control over light's path using curved reflecting surfaces.

What is Refraction?

Refraction is the bending of light as it passes between media with different refractive indices. When light travels from air into glass, it slows down. The speed of light in a medium is c/n, where c is light's speed in vacuum and n is the refractive index. Water has n ≈ 1.33, glass typically n ≈ 1.5, and diamond n ≈ 2.4. Because light slows in the denser medium, its direction changes at the boundary—it bends toward the normal when entering a denser medium and away from the normal when entering a less dense medium.

Snell's law quantifies refraction: n₁sin(θ₁) = n₂sin(θ₂), where n is refractive index and θ is angle from the normal. This relationship reveals why objects underwater appear displaced from their true position. Light from a submerged fish reaches your eye by refracting at the water-air interface, bending away from the normal. Your brain assumes light traveled in straight lines, placing the fish closer and shallower than it actually is. This illusion stems from refraction's geometric consequences.

Refraction arises from light's interaction with atoms and molecules in materials. Electromagnetic waves cause electrons to oscillate. These oscillating electrons absorb energy from the incident wave and re-radiate it, creating a secondary wave that interferes with the incident wave. In transparent materials, this interference modifies the wave's speed without significant energy absorption. The refractive index fundamentally reflects the electronic structure of the material—materials with easily polarizable electrons have higher refractive indices.

Dispersion occurs because the refractive index depends on wavelength. Blue light bends more than red light in glass because blue light's higher frequency causes stronger interaction with electrons. This wavelength dependence explains why prisms separate white light into rainbows and why chromatic aberration affects uncompensated lenses. Different wavelengths reach different focal points, creating color fringes. Telescope and microscope designers must correct for dispersion using multiple lens elements of different materials.

The Mathematics of Reflection and Refraction

Total Internal Reflection

Total internal reflection is a remarkable phenomenon where light traveling through a denser medium cannot escape to a less dense medium at large angles. Instead of refracting out, all light reflects back. This occurs when the refraction angle would exceed 90 degrees—Snell's law cannot be satisfied, so the light must reflect instead.

Consider light in glass traveling toward the air boundary. At small incident angles, light partially reflects and partially refracts out. As the incident angle increases, the refracted angle increases faster (because of Snell's law). At the critical angle θc = arcsin(n_air/n_glass) = arcsin(2/3) ≈ 41.8° for typical glass, the refracted angle reaches 90°. Light travels along the interface. For incident angles exceeding the critical angle, no refracted ray exists—total internal reflection occurs.

Total internal reflection has profound applications. Diamond's high refractive index (n ≈ 2.4) produces a critical angle of only about 24°. This small critical angle means most light entering a diamond undergoes total internal reflection, bouncing inside and exiting only through the top. This brilliant sparkling is the source of diamond's beauty. Optical fibers exploit total internal reflection to guide light thousands of kilometers with minimal loss, enabling global telecommunications.

Prisms use total internal reflection to redirect light. A 45-90-45 degree prism can reflect light by 90 degrees or separate it by color. Binoculars and periscopes employ prisms instead of mirrors because mirrors lose light through incomplete reflection, while prisms with total internal reflection reflect nearly all light. This ensures bright, clear images despite multiple reflections.

Fiber Optics and Telecommunications

Optical fibers are among humanity's most important inventions, enabling the global internet. A typical optical fiber has a core diameter of 8-10 micrometers (about 1/10 the width of human hair) surrounded by cladding of slightly lower refractive index. Light entering the fiber at shallow angles repeatedly undergoes total internal reflection, bouncing along the core toward its destination.

Single-mode fibers have cores so small (around 8 micrometers) that only one path for light (one "mode") can propagate—the fundamental mode. Light follows essentially straight paths with little interference. These fibers enable long-distance communication because light signals maintain quality over thousands of kilometers. Multi-mode fibers have larger cores (50-62 micrometers), allowing many light paths. Because different paths have different lengths, multi-mode fibers suffer modal dispersion—different path delays blur signals. Multi-mode fibers excel for short distances, like within buildings.

Attenuation—loss of light intensity—determines how far signals can travel. Modern optical fibers lose only about 0.2 dB per kilometer at telecommunications wavelengths near 1550 nanometers. This remarkably low loss contrasts with copper cables (hundreds of dB per kilometer) and explains why every long-distance communication link uses fiber. At 0.2 dB/km, light can travel 100 kilometers and still retain half its power. Submarine cables spanning continents use fiber, with repeaters every 50-100 kilometers to amplify signals.

Chromatic dispersion limits bit rates in fiber. Different wavelengths travel at slightly different speeds due to material dispersion (refractive index depends on wavelength) and waveguide dispersion (fiber geometry affects different wavelengths differently). After long distances, a brief light pulse broadens as its wavelength components arrive at different times, eventually blending with subsequent pulses. This limits communication speed. Compensation techniques include using dispersion-shifted fibers designed to minimize dispersion at telecommunications wavelengths, or electronic equalization at the receiver.

Mirages, Prisms, and Applications

A mirage is a refraction phenomenon caused by temperature gradients near the ground. Hot sand creates a layer of hot air near the surface, which has lower refractive index than cooler air above. Light from the sky traveling downward refracts repeatedly as it passes through these layers. At a sufficiently shallow angle, the light undergoes total internal reflection from the hot air layer and travels back upward to the observer's eye. The observer sees an inverted image of the sky appearing as a reflection from the ground, creating the illusion of water—hence the term "mirage."

Prisms are optical elements exploiting refraction and total internal reflection. A 45-45-90 degree glass prism can redirect light by 90 degrees through total internal reflection, acting as a mirror without the losses of reflective coatings. A 60-60-60 degree prism or equilateral prism disperses light by wavelength—different colors refract at slightly different angles, separating white light into a spectrum. This chromatic dispersion occurs because the refractive index increases slightly for shorter wavelengths.

The minimum deviation condition for prism dispersion occurs when light travels symmetrically through the prism—entering and exiting at equal angles to the respective surfaces. At this angle, dispersion is maximum, producing the brightest, most saturated spectrum. Newton's prism experiments demonstrating that white light consists of colors used minimum deviation geometry. Modern spectrometers and colorimeters employ prisms exploiting dispersion.

Anti-reflective coatings use thin films to reduce reflection. A quarter-wave coating of low-refractive-index material on a high-index surface causes destructive interference between light reflecting from the coating's top and bottom surfaces, canceling reflection. Coated optics appear slightly greenish (the reflected green light is suppressed) and transmit more light. Cameras, telescopes, and eyeglasses all use anti-reflective coatings to improve light transmission and reduce glare.