Total Internal Reflection Calculator
Enter two refractive indices and the incidence angle to compute the critical angle θc=arcsin(n₂/n₁) and whether total internal reflection occurs. Glass n₁=1.5 to air n₂=1.0 → θc=41.8°; at 45° it fully reflects.
Input Data
Results
At a glance:Total internal reflection (TIR): when light travels from a denser medium (large n₁) into a rarer one (small n₂) and the incidence angle θ exceeds the critical angle θc, all the light is reflected back into the denser medium with no refracted ray. The critical angle is θc=arcsin(n₂/n₁) (valid only for n₁>n₂). For θ<θc part refracts and part reflects; at θ=θc the refracted ray runs along the interface (refracted angle =90°). If n₁<n₂ there is no TIR (critical angle 90°). Applications: (1) optical fibre (TIR carries light signals); (2) right-angle prisms (90° reflection, binoculars); (3) bicycle reflectors (cat's eye); (4) diamonds (high n=2.42, small critical angle, lots of TIR, sparkle); (5) mirages (atmospheric TIR).
Formula
Critical angle: θc = arcsin(n₂/n₁) (n₁ > n₂)
Snell's law: n₁·sinθ₁ = n₂·sinθ₂
TIR condition: θ₁ > θc
At critical: refracted angle = 90°
$$\theta_c = \arcsin\left(\frac{n_2}{n_1}\right), \quad n_1 > n_2, \quad \text{TIR when } \theta_1 > \theta_c$$How to Use
- Enter the denser medium index n₁.
- Enter the rarer medium index n₂.
- Enter the incidence angle θ (°).
- The tool gives the critical angle, refracted angle and whether TIR occurs.
Case Studies
Optical fibre and prisms
Fibre: core n=1.5, cladding n=1.46 → θc=arcsin(1.46/1.5)=76.7°. Light TIR-propagates in the core, bendable. Telecom, endoscopes.
Right-angle prism: glass n=1.5, incidence 45° > θc 41.8° → TIR. Turns the light 90°. Two prisms erect the image in binoculars.
Bicycle reflector: arrays of plastic micro-prisms reflect light back from the rear via TIR. Cat's-eye, night safety.
Diamonds and mirages
Diamond: n=2.42, θc=24.4°. Most incident light TIRs internally, exits after multiple cut facets — hence the sparkle. Precise cut angles guarantee TIR.
Mirage: lower air is hot (low density, low n), upper air cold (high density, high n). Light curves up and TIRs, the ground shows a sky-like water surface. Common in deserts and on highways.
Underwater periscope: water-air TIR angle 48.8°. The underwater TIR zone compresses the sky into a 97.6° cone (Snell's window).
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.