Angle of Incidence Calculator (Snell's Law)
Enter the two optical media (or their refractive indices) and one angle to solve for the other using Snell's Law. The calculator also finds the critical angle for total internal reflection, the Brewster polarisation angle, and flags when total internal reflection occurs. Switch between solving for the angle of incidence or the angle of refraction, and step through the worked math below the result.
Formula
Worked example
Light travels from air (n₁ = 1.000273) into crown glass (n₂ = 1.520) and the refraction angle is 30°. By Snell's Law: sin(θ₁) = (1.520 / 1.000273) × sin(30°) = 1.5196 × 0.5 = 0.7598, so θ₁ = arcsin(0.7598) ≈ 49.5°. The Brewster angle is arctan(1.520 / 1.000273) ≈ 56.6°. Total internal reflection cannot occur here because n₁ < n₂.
What is the angle of incidence?
The angle of incidence is the angle between an incoming ray of light (or any wave) and the normal to the surface at the point where the ray strikes. The normal is an imaginary line perpendicular to the surface. When light crosses from one transparent medium into another, both the angle of incidence (θ₁) and the angle of refraction (θ₂) are measured from this normal, not from the surface itself. If the ray hits the surface head-on (perpendicular), both angles are zero and the ray passes straight through without bending.
Snell's Law: how it works
Snell's Law, also called the law of refraction, states that n₁ sin(θ₁) = n₂ sin(θ₂), where n₁ and n₂ are the refractive indices of the two media. The refractive index of a material is the ratio of the speed of light in a vacuum to the speed of light in that material. A higher refractive index means light travels more slowly in that medium. When light moves from a lower-index medium into a higher-index medium (for example from air into glass), it slows down and bends toward the normal, so the refraction angle is smaller than the incidence angle. The reverse happens when light exits a denser medium: it speeds up and bends away from the normal.
Critical angle and total internal reflection
When light travels from a denser medium (higher n) into a less dense medium (lower n), there is a special angle called the critical angle (θ_c) beyond which no refracted ray can exist. At this angle sin(θ₂) would have to equal or exceed 1, which is impossible. The result is total internal reflection: all the light bounces back into the denser medium as if the interface were a perfect mirror. The critical angle is θ_c = arcsin(n₂ / n₁). Optical fibres rely entirely on this principle: a thin glass core surrounded by cladding of slightly lower refractive index traps light and guides it around bends over kilometres with minimal loss.
Brewster's angle and polarisation
Brewster's angle (θ_B) is the incidence angle at which reflected light becomes completely polarised. At this angle, the reflected and refracted rays are perpendicular to each other, which means the p-polarised component (oscillating in the plane of incidence) cannot be reflected. Brewster's angle is calculated as θ_B = arctan(n₂ / n₁). For air-to-glass this is roughly 56-57°. Photographers use this: tilting a polarising filter to Brewster's angle eliminates glare from water or glass surfaces. Anti-reflection coatings in optical instruments are also designed around this principle.
Real-world applications
The angle of incidence and Snell's Law appear wherever light crosses a boundary. Eyeglass lens designers use them to specify how much a lens bends light toward the focal point. Prism-based spectrometers separate white light into colours because different wavelengths are refracted by slightly different amounts (dispersion). Laser safety engineers calculate beam paths through protective windows. Solar panel designers tilt panels at the optimal angle of incidence for the sun's position to maximise power generation. In medical endoscopy, light is delivered to tissue through flexible optical fibre bundles kept at angles below the critical angle.
Common optical media and refractive indices
| Medium | Refractive index (n) | Notes |
|---|---|---|
| Vacuum | 1.000 000 | Exact by definition |
| Air (STP) | 1.000 273 | Standard temperature and pressure |
| Ice | 1.310 | Water H₂O in solid phase |
| Water (20 °C) | 1.333 | Pure liquid water |
| Ethanol | 1.361 | Common laboratory solvent |
| Glycerol | 1.473 | Dense, viscous liquid |
| Acrylic (PMMA) | 1.490 | Plexiglass, perspex |
| Oil (mineral) | 1.515 | Immersion oil for microscopy |
| Crown glass | 1.520 | Standard optical glass |
| Quartz (fused) | 1.544 | Silicon dioxide glass |
| Flint glass | 1.620 | High-dispersion optical glass |
| Diamond | 2.420 | Highest common natural index |
Refractive indices at visible light wavelengths (~589 nm, sodium D line). Values vary slightly with wavelength and temperature.
Frequently asked questions
What is the difference between angle of incidence and angle of refraction?
The angle of incidence (θ₁) is the angle the incoming ray makes with the normal to the surface in the first medium. The angle of refraction (θ₂) is the angle the transmitted ray makes with the normal on the other side of the interface in the second medium. Both are always measured from the normal, not from the surface. Snell's Law links them: n₁ sin(θ₁) = n₂ sin(θ₂).
What is the refractive index of air?
The refractive index of air at standard temperature (0 °C) and pressure (1 atm) is approximately 1.000273. It is so close to 1 (the vacuum value) that for most practical calculations air is treated as a vacuum with n = 1. The value varies slightly with temperature, pressure, humidity, and wavelength.
When does total internal reflection occur?
Total internal reflection occurs when two conditions are met: (1) the light is travelling from an optically denser medium into a less dense one (n₁ > n₂), and (2) the angle of incidence equals or exceeds the critical angle θ_c = arcsin(n₂ / n₁). Below the critical angle, a refracted ray still exits. At or above it, all light reflects back into the first medium. This is how optical fibres and diamond facets work.
How do I find the angle of incidence if I know the angle of refraction?
Use Snell's Law rearranged: sin(θ₁) = (n₂ / n₁) × sin(θ₂), then take the arcsine. For example, if light refracts at 30° in glass (n = 1.52) entering from air (n = 1.000273), then sin(θ₁) = (1.52 / 1.000273) × sin(30°) = 1.5196 × 0.5 = 0.7598, giving θ₁ = arcsin(0.7598) ≈ 49.5°. This calculator does this in one step.
What is Brewster's angle used for?
Brewster's angle is the incidence angle at which reflected light is completely polarised. Photographers use polarising filters at this angle to remove glare from non-metallic surfaces like water and glass, revealing colours underneath. Engineers design anti-reflection coatings using this principle, and laser cavities use Brewster windows to produce polarised output with minimal reflection loss.
Does the angle of incidence equal the angle of reflection?
Yes, for specular (mirror-like) reflection on a smooth surface, the angle of incidence always equals the angle of reflection. This is the law of reflection, and it holds regardless of the refractive indices of the media. It is separate from Snell's Law, which governs the transmitted (refracted) ray that continues into the second medium.