Snell's Law & Refraction Calculator

Snell's Law Guide

What's the key thing to understand about Snell's Law?

Snell's Law describes how light bends when passing between media with different refractive indices: n₁ sin θ₁ = n₂ sin θ₂. The refractive index n = c/v, where c is the speed of light in vacuum and v is the speed in the medium. Light bends toward the normal when entering a denser medium (higher n) and away from the normal when entering a less dense medium. At normal incidence (θ₁ = 0°), there is no bending regardless of the refractive indices.

What do I need to know about Total Internal Reflection?

When light travels from a denser medium (higher n) to a less dense one (lower n), there is a critical angle above which all light is reflected back into the denser medium — no refraction occurs. Critical angle θc = arcsin(n₂/n₁). For glass (n=1.5) to air: θc = arcsin(1.0/1.5) = 41.8°. For diamond (n=2.417) to air: θc = 24.4° — this is why diamonds sparkle so brilliantly, trapping and internally reflecting light from many angles.

What do I need to know about Optical Fibres?

Optical fibre communication relies entirely on total internal reflection. Light travels down a glass fibre and is totally internally reflected at every interface with the lower-index cladding surrounding the core. This allows light to travel around bends and over thousands of kilometres with minimal loss. The core has a slightly higher refractive index than the cladding. Modern fibre optic cables transmit data at near the speed of light with bandwidth far exceeding copper cables — they underpin the entire modern internet backbone.

What do I need to know about Everyday Refraction?

Why a straw appears bent in water: light bends as it crosses the water-air interface, shifting the apparent position of the submerged part. Why swimming pools look shallower than they are: upward-moving light bends away from the normal at the water-air interface, making the bottom appear closer. Mirages: hot road surfaces create layers of different-temperature air with different refractive indices, causing total internal reflection of skylight to appear as a reflection on the road surface.

What should I know about the Refractive Index of Common Materials?

The refractive index (n) of a material determines how much it bends light, and is defined as the ratio of the speed of light in a vacuum to the speed of light within that material: n = c/v. Common approximate values: a vacuum has n = 1 exactly, by definition; air is very close to this at n ≈ 1.0003 (often rounded to 1.00 for everyday calculations); water has n ≈ 1.33; ordinary crown glass used in windows and lenses has n ≈ 1.52; and diamond has one of the highest refractive indices of any natural material at n ≈ 2.42, which is a major reason diamonds appear so brilliant — light entering a diamond bends sharply and tends to undergo extensive total internal reflection before exiting, creating the sparkle and 'fire' that gemstone cutters design facets to maximise. The larger the difference in refractive index between two materials at a boundary, the more dramatically light bends when crossing between them, which is exactly why a straw looks only slightly displaced in water (n=1.33) but a diamond's internal light paths bend so sharply.

What's the key thing to understand about Dispersion and Why Prisms Split Light?

A material's refractive index isn't actually a single fixed number — it varies slightly depending on the wavelength (colour) of light passing through it, a phenomenon called dispersion. Violet and blue light (shorter wavelengths) are refracted slightly more strongly than red and orange light (longer wavelengths) in most transparent materials, including ordinary glass. This is exactly why a glass prism splits white light into a visible rainbow spectrum: as white light (a mixture of all visible wavelengths) enters and exits the angled prism surfaces, each colour bends by a very slightly different amount according to Snell's Law, causing the colours to separate and fan out by the time they emerge. The same physical principle, occurring naturally in raindrops rather than glass, produces a rainbow — sunlight refracts and internally reflects within each spherical raindrop, and dispersion separates the white sunlight into its component colours, which is why a rainbow always has red on the outer edge (bent least) and violet on the inner edge (bent most) when viewed with the sun behind you.

What should I know about a Worked Snell's Law Example?

Light travelling from air (n₁ = 1.00) into glass (n₂ = 1.50) at an angle of incidence of 30° refracts according to n₁ sinθ₁ = n₂ sinθ₂: sinθ₂ = (1.00 × sin30°)/1.50 = 0.5/1.50 = 0.333, giving θ₂ ≈ 19.5°. The ray bends toward the normal because it's entering a denser (higher-index) medium and slowing down — exactly the same relationship, worked in reverse, that lets a critical angle and total internal reflection be found when light tries to exit back into the less dense medium at a steep enough angle.

Critical Angle for Common Material Pairs

The critical angle is the angle of incidence (inside the denser medium) at which the refracted ray grazes exactly along the boundary at 90°, marking the onset of total internal reflection for any steeper angle. It's found with sinθc = n₂/n₁, where n₁ is the denser medium the light is traveling in and n₂ is the less dense medium it's trying to exit into.

The diamond row explains why diamonds sparkle so intensely: because the critical angle is so small (about 24°), almost any light entering the stone hits the internal facets at angles steeper than that and reflects internally multiple times before escaping, rather than passing straight through.

Brewster's Angle and Polarization by Reflection

Besides the critical angle, there's a second special angle in reflection and refraction worth knowing: Brewster's angle, the angle of incidence at which reflected light becomes completely polarized parallel to the surface. It's found with tanθB = n₂/n₁. For light reflecting off water (n = 1.33) from air (n = 1.00), θB = arctan(1.33) ≈ 53.1°. At that specific angle, the reflected glare is fully polarized, which is exactly why polarized sunglasses are so effective at cutting reflected glare off water, snow, and wet roads — they block the horizontally polarized component that dominates at near-Brewster viewing angles. For light hitting ordinary glass (n ≈ 1.52), Brewster's angle is arctan(1.52) ≈ 56.7°. Photographers use the same principle deliberately: tilting a polarizing filter and viewing a reflective surface near its Brewster angle lets them dial out reflections on water or glass almost entirely, since that's the viewing angle where the reflected glare is most purely polarized and most easily filtered out.

Worked Example: Light Entering a Swimming Pool from Air

Sunlight strikes the surface of a swimming pool at an angle of incidence of 40° from the vertical (the normal). Using Snell's law with n₁ = 1.00 (air) and n₂ = 1.33 (water): sinθ₂ = (n₁ sinθ₁)/n₂ = (1.00 × sin40°)/1.33 = 0.643/1.33 ≈ 0.483, so θ₂ = arcsin(0.483) ≈ 28.9°. The light ray bends toward the normal by more than 11°, which is why objects viewed underwater from above appear shifted from where they actually are, and why a straight stick dipped at an angle into a pool appears to bend sharply at the water's surface. This same bending is why swimming pools and lakes always look shallower than they really are when viewed from directly above the surface — the apparent depth is less than the true depth by a factor related to the ratio of the refractive indices, roughly apparent depth ≈ true depth × (n_air/n_water) ≈ true depth × 0.75 for near-vertical viewing.

Table: Refractive Index and Speed of Light in Common Media

Refractive index n is defined as the ratio of light's speed in a vacuum to its speed in the medium: n = c/v. A higher index means light travels proportionally slower through that material.