Total Internal Reflection and Fiber Optics

Total internal reflection (TIR) occurs when light traveling in a denser medium strikes the boundary with a less dense medium at an angle of incidence greater than the critical angle θ_c, where sin θ_c = n₂ ÷ n₁, and all the light is reflected back; optical fibers use TIR to guide light over long distances. In brief: two conditions are needed: light must go from a higher to a lower refractive index (glass to air), and the angle of incidence must exceed the critical angle. For glass-air (n = 1.50), θ_c = 41.8°; for water-air, 48.8°; for diamond-air, 24.4°.

The sparkle of a diamond, the shimmering underside of the water surface seen by a diver and the internet cables under the oceans all depend on total internal reflection. This guide explains the critical angle and the conditions for TIR, describes how optical fibers and prisms work, compares media in a table, works through calculations for the critical angle and a fiber, and corrects common mistakes.

What is total internal reflection?

When light passes from glass into air, it bends away from the normal. As the angle of incidence increases, the refracted ray gets closer to the boundary, until at the critical angle θ_c the refracted ray travels along the surface (angle of refraction 90°). For larger angles of incidence, there is no refracted ray at all: all the light is reflected back into the glass according to the law of reflection, and the boundary acts as a perfect mirror. This is TIR. It does not occur when light goes from a less dense to a denser medium, such as from air into glass.

How is the critical angle found?

From Snell’s law, n₁ sin θ_c = n₂ sin 90°, so sin θ_c = n₂ ÷ n₁ (with n₁ > n₂). For a boundary with air, n₂ ≈ 1, so sin θ_c = 1 ÷ n. The higher the refractive index, the smaller the critical angle and the easier it is to get TIR. Applications:

Prisms in binoculars and periscopes: a right-angle glass prism reflects light by TIR at 45° (greater than 41.8°), more efficiently than silvered mirrors, and it does not tarnish.
Diamonds (n = 2.42, θ_c = 24.4°) trap light by repeated TIR, and the facets are cut so that light exits the top face, giving brilliance.
Reflectors on bicycles and road signs use corner prisms.
Mirages and the apparent mirror surface under water seen from below.
Medical endoscopes use bundles of fibers to see inside the body.

Rays in glass (n = 1.50) reaching a glass-air boundary. At 30°, less than the critical angle, the ray refracts out and bends away from the normal to 48.6°. At the critical angle, 41.8°, the refracted ray skims along the surface (dashed). At 55°, greater than the critical angle, all the light is reflected back into the glass: total internal reflection.

How do optical fibers work?

An optical fiber is a thin strand of glass or plastic with a core of refractive index n_core surrounded by a cladding of slightly lower index n_clad. Light entering the end of the core at a small angle to the axis hits the core-cladding boundary at an angle larger than the critical angle, and it is totally reflected again and again, so that it travels along the fiber, even around bends. Pulses of infrared light carry digital signals. Compared with copper wires, fibers have lower signal loss (modern fibers lose about 0.2 dB/km), carry much more information, are not affected by electrical interference, and are lightweight. Modern fiber cables, including submarine cables across the oceans, carry most of the world’s data traffic. The numerical aperture, NA = √(n_core² − n_clad²), gives the maximum angle (the acceptance angle) at which light can enter from air and still be guided: sin θ_max = NA.

Critical angles compared

Critical angles for light going into air
Material	Refractive index	Critical angle
Water	1.33	48.8°
Glass (crown)	1.50	41.8°
Diamond	2.42	24.4°

Worked example: critical angles and a fiber

Glass to air. sin θ_c = 1 ÷ 1.50 = 0.667, so θ_c = 41.8°.
Glass to water. sin θ_c = 1.33 ÷ 1.50 = 0.887, so θ_c = 62.5°, larger because the indices are closer.
Prism. A ray enters one face of a 45°–45°–90° glass prism (n = 1.50) at normal incidence and strikes the long face at 45°. Since 45° > 41.8°, TIR occurs and the ray turns through 90°. If the prism were in water, θ_c = 62.5° > 45°, and the light would pass through the face.
Optical fiber. A fiber with n_core = 1.48 and n_clad = 1.46: sin θ_c = 1.46 ÷ 1.48 = 0.9865, so θ_c = 80.6°. Rays that hit the core-cladding boundary at more than 80.6° from the normal (that is, less than 9.4° from the axis) are guided.
Acceptance angle. NA = √(1.48² − 1.46²) = √(2.1904 − 2.1316) = √0.0588 = 0.243, so the maximum angle of entry from air is θ_max = sin⁻¹(0.243) = 14.0°.

How was TIR discovered and used?

Johannes Kepler described total internal reflection in 1611. Daniel Colladon and Jacques Babinet demonstrated in the 1840s that light can be guided in a jet of water. John Tyndall repeated the demonstration for the Royal Society in 1854. Narinder Singh Kapany developed fiber optics in the 1950s, and Charles Kao showed in 1966 that glass fibers of high purity could carry signals over long distances, for which he shared the 2009 Nobel Prize in Physics. Low-loss fibers were made by Corning in 1970.

Which mistakes are common?

Mistake 1: Expecting TIR from a less dense to a denser medium

There is no critical angle when light goes from air into glass. TIR requires n₁ > n₂.

Mistake 2: Using an angle less than the critical angle

At angles below θ_c, some of the light is refracted out. TIR requires the angle to be larger.

Mistake 3: Measuring the angle from the surface

θ_c is measured from the normal.

Mistake 4: Thinking that TIR is a refraction effect

It is a reflection: no light is transmitted (apart from a thin evanescent field).