Total internal reflection is the complete reflection of light back into the material it is traveling through. It occurs only when light reaches a boundary from a higher-refractive-index medium toward a lower-index one, at an angle greater than the critical angle.
What total internal reflection means
When light reaches the boundary between two materials, it can reflect, refract, or do both. In total internal reflection (TIR), no light travels away as a propagating refracted ray in the second material; instead, the light is reflected back into the first material. The effect is used to guide or redirect light, including in optical fibers and reflective prisms.
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The angles are measured from the normal: an imaginary line perpendicular to the boundary at the point where the light hits it. The angle between the incoming ray and the normal is the angle of incidence.
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- Light must travel from higher refractive index to lower refractive index. If the first medium has index n₁ and the second has index n₂, then n₁ > n₂.
- The angle of incidence must exceed the critical angle. Both angles are measured from the normal.
Both conditions are essential: light does not undergo TIR simply because it strikes a boundary at a steep angle. The direction of travel through the materials matters too. OpenStax’s explanation of total internal reflection describes these conditions and their applications.
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How to calculate the critical angle
The critical angle, θc, is the incidence angle at which the refracted ray travels along the boundary. Snell’s law is n₁ sin θ₁ = n₂ sin θ₂. At the critical angle, θ₂ = 90°, so sin θ₂ = 1. Substituting and solving gives:
θc = sin⁻¹(n₂/n₁), for n₁ > n₂
Use the refractive indices for the two materials under the relevant light conditions. The ratio n₂/n₁ determines the critical angle: for a given higher-index material, a lower-index adjoining material gives a smaller ratio and therefore a smaller critical angle. The formula and angle convention follow from Snell’s law and refraction.
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What happens below, at, and above the critical angle
- Below θc: Some light refracts into the second medium, while some reflects.
- At θc: The refracted ray runs along the boundary, at 90° to the normal. This is the limiting case, not total internal reflection as defined by an incidence angle greater than θc.
- Above θc: In the elementary ray description, there is no propagating refracted ray into the second medium; the light is reflected back into the first.
Example: water meeting air
For a water–air boundary, HyperPhysics gives a critical angle of 48.6°. Treat that as a reference example, not a universal value for every water–air setup: the calculation depends on the refractive indices used and the light conditions. HyperPhysics explains the water–air example.
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Where total internal reflection is used
Optical fibers and waveguides guide light
An optical fiber can guide light when its higher-index core is surrounded by lower-index material. Light that meets the core boundary at suitable angles is reflected back into the core, allowing it to travel along the fiber. Fiber optics carry communications signals; image-forming fibers are also used in endoscopes. OpenStax notes that “Fiber optics is one application of total internal reflection that is in wide use.” OpenStax, University Physics Volume 3, section 1.4.
Prisms redirect light
Reflective prisms can use TIR to redirect light inside an optical instrument. OpenStax describes prism reflections in binoculars and identifies periscopes as another application. Here the aim is to change the light’s direction within an instrument, rather than carry it along a flexible path as a fiber does. OpenStax’s College Physics 2e discussion covers these examples.
Diamonds and sparkle
Diamond’s high refractive index contributes to internal reflection and sparkle. TIR is one part of the optical effect, not a complete explanation of every aspect of a diamond’s appearance.
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