Total Internal Reflection
Total Internal Reflection (TIR) occurs when a light ray travelling through a denser medium strikes the boundary with a less dense medium at an angle greater than the critical angle, causing the ray to be completely reflected back into the denser medium rather than refracted out. The critical angle θ_c is defined as sin θ_c = n₂/n₁, where n₁ > n₂. TIR is the operating principle behind optical fibres, diamonds' brilliance, and binocular prisms.
Key Formula
sin(θ_c) = n2 / n1
LaTeX: \sin\theta_c = \frac{n_2}{n_1}
| Symbol | Meaning | Unit |
|---|---|---|
| θ_c | Critical angle | degrees |
| n₁ | Refractive index of denser medium (incident) | dimensionless |
| n₂ | Refractive index of less dense medium (second) | dimensionless |
Worked Example
Problem
Calculate the critical angle for a glass–air interface where the glass has a refractive index of 1.50 and air has n = 1.00.
Solution
Step 1: Use the critical angle formula: sin θ_c = n₂ / n₁ Step 2: Substitute: sin θ_c = 1.00 / 1.50 = 0.6667 Step 3: θ_c = arcsin(0.6667) ≈ 41.8° Step 4: Any ray hitting the glass–air boundary at an angle > 41.8° undergoes TIR.
Answer
Critical angle ≈ 41.8°; rays beyond this angle are totally internally reflected.
Critical Angles for Common Medium–Air Interfaces
| Denser Medium | Refractive Index (n) | Critical Angle (°) | Application |
|---|---|---|---|
| Water | 1.333 | 48.8 | Underwater photography shimmer |
| Crown Glass | 1.520 | 41.1 | Optical fibre cladding |
| Dense Flint Glass | 1.700 | 36.0 | Prism binoculars |
| Diamond | 2.417 | 24.4 | Gemstone brilliance |
| Optical Fibre Core | 1.460 | 43.2 | Telecommunications |
Interactive Tools
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Related Terms
Snell's Law
Snell's Law (also called the law of refraction) states that the product of the refractive index and the sine of the angle of incidence is constant across the interface between two media: n₁ sin θ₁ = n₂ sin θ₂. It quantitatively describes how a light ray changes direction when it transitions between media of different optical densities. Snell's Law is the cornerstone of lens design, fibre optic engineering, and the correction of refractive vision errors.
Light Refraction
Light refraction is the bending of a light ray as it passes from one transparent medium into another of different optical density, caused by a change in the wave's speed. The greater the difference in refractive indices between the two media, the more the ray bends toward or away from the normal. Refraction is responsible for phenomena such as the apparent bending of a straw in water, the formation of rainbows, and the focusing action of lenses.
Optical Dispersion
Optical dispersion is the phenomenon in which the refractive index of a medium varies with the wavelength (frequency) of light, causing different colours to refract by different amounts and thereby separate from one another when passing through a dispersive medium such as a glass prism or water droplet. Shorter wavelengths (violet) are refracted more than longer wavelengths (red) in normal dispersion. Dispersion is responsible for the formation of rainbows, chromatic aberration in lenses, and the spectral analysis of light sources.
The word "total" specifies completeness of reflection; "internal" indicates the ray is inside the denser medium. The phenomenon was first explained scientifically by Johannes Kepler in 1611 in "Dioptrice". The term "total internal reflection" became standard in optics texts during the 19th century following systematic work by Augustin-Jean Fresnel.