concept

total internal reflection

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.
6 related articles · 5 formulas

Description

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.

History

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.

How it works

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.

In practice

Fiber-optic communication (internet), prisms in binoculars, medical endoscopes, touch sensors in screens.

Fun fact

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.

Concept graph →

Formulas

$$N = \frac{I_{\text{min}}}{I_{\text{max}}}$$
Null depth is the ratio of minimum to maximum intensity.
What is in the formula
Nvariablenull depth ratio — null depth ratio
I_minvariableminimum intensity — intensity watt per square metre
I_maxvariablemaximum intensity — intensity watt per square metre
n=\frac{i_{{min}}}{i_{{max}}} · 2607.26168v1
$$c_{\rm app} = c_{\rm int} + E$$
Apparent color equals intrinsic color plus color excess.
What is in the formula
c_appvariableApparent color index of a celestial object — color index
c_intvariableIntrinsic color index of the object — color index
EvariableColor excess due to interstellar extinction — color excess
c_{\rmapp}=c_{\rmint}+e · 2606.06593v1
$$\tau = \int_{0}^{\infty} C(t) \, dt$$
The correlation time equals the integral of the autocorrelation function.
What is in the formula
\tauvariableintegral autocorrelation time — autocorrelation time second
C(t)variableautocorrelation function — autocorrelation function
tvariabletime lag — time second
\int_{0}^{\infty}operatorintegration from zero to infinity — definite integral
τ_{γ}=\int_{0}^{\infty}c_{γ}(t)dt · 2606.31185
$$|t|^2 + |r|^2 = 1$$
The sum of transmitted and reflected intensities equals the incident intensity.
What is in the formula
tvariabletransmission amplitude — transmission amplitude
rvariablereflection amplitude — reflection amplitude
|t|^2+|r|^2=1 · 2605.11065v1
$$f(x) = x$$
The identity map returns each input unchanged.
What is in the formula
xvariableinput value
f(x)variableoutput value
r_z(x)=x · 2607.01429

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