Total internal reflection is the phenomenon where light is completely reflected at a boundary between two media when the angle of incidence exceeds a critical value.Total internal reflection (TIR) occurs when light tries to go from a denser optical medium (with higher refractive index n₁) to a less dense one (n₂ < n₁) at an angle greater than the critical one. In this case, it is completely reflected back, and only an evanescent field remains beyond the boundary. This effect is indispensable in waveguides.
The first description of total internal reflection was given by Kepler in 'Dioptrice' (1611). In 1621, Snellius discovered the law of refraction, from which the critical angle is directly derived. In the 1820s, Fresnel developed the theory, including evanescent waves. In 1854, Tyndall demonstrated a light guide using a water jet - a prototype of optical fiber. In the 1960s, practical application of fiber optics began.
Condition for TIR: n₁ > n₂ and angle of incidence θ₁ > θ_c. Conservation of the tangential component of the wave vector gives k_x = (n₁ ω/c) sin θ₁. When θ₁ > θ_c, k_x exceeds n₂ ω/c, so the normal component k_z₂ becomes imaginary: k_z₂ = iκ. The field decays exponentially: exp(−κ z), penetration depth δ = 1/κ = λ₀/(2π n₁ √(sin²θ₁ − (n₂/n₁)²)). The evanescent wave carries energy along the boundary but not into the volume. The phase shift upon reflection differs for s- and p-polarizations. Frustrated TIR is analogous to quantum tunneling: when two prisms are brought within a distance of about δ, light partially penetrates into the second.
Fiber-optic communication (internet), prisms in binoculars, medical endoscopes, touch sensors in screens.
The brilliance of diamonds is a result of total internal reflection: due to the high refractive index (2.42), light repeatedly reflects off facets and exits at various angles, creating a play of highlights.
| N | variable | null depth ratio — null depth ratio |
| I_min | variable | minimum intensity — intensity watt per square metre |
| I_max | variable | maximum intensity — intensity watt per square metre |
| c_app | variable | Apparent color index of a celestial object — color index |
| c_int | variable | Intrinsic color index of the object — color index |
| E | variable | Color excess due to interstellar extinction — color excess |
| \tau | variable | integral autocorrelation time — autocorrelation time second |
| C(t) | variable | autocorrelation function — autocorrelation function |
| t | variable | time lag — time second |
| \int_{0}^{\infty} | operator | integration from zero to infinity — definite integral |
| t | variable | transmission amplitude — transmission amplitude |
| r | variable | reflection amplitude — reflection amplitude |
| x | variable | input value |
| f(x) | variable | output value |






