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 and θ₁ and θ₂ are the angles the light ray makes with the normal to the interface. It was derived geometrically by Willebrord Snellius around 1621 and published by René Descartes in 1637. It follows from the wave nature of light and the requirement that the tangential component of the wave vector is continuous across an interface.

What is the refractive index?

The refractive index n of a medium is the ratio of the speed of light in vacuum (c ≈ 2.998 × 10⁸ m/s) to the phase velocity of light in that medium: n = c / v. Because light slows down in matter, n ≥ 1 for all ordinary materials (vacuum n = 1 exactly). Higher n means greater optical density: diamond (n ≈ 2.417) bends light far more sharply than glass (n ≈ 1.52), which is why diamonds sparkle. The values in this calculator are for ~589 nm (the sodium D-line), a standard reference wavelength.

What is total internal reflection and when does it happen?

Total internal reflection (TIR) occurs when light tries to pass from a denser medium (higher n) to a rarer medium (lower n) at an angle equal to or greater than the critical angle θ_c = arcsin(n₂ / n₁). At and beyond θ_c, sin θ₁ > n₂/n₁, so Snell's law would require sin θ₂ > 1 — which is impossible. Instead, 100 % of the light is reflected back into the first medium. TIR is the operating principle of optical fibres, retroreflectors, and diamond cuts. Water-to-air TIR starts at θ_c ≈ 48.75°.

What is the critical angle for glass to air?

For Crown glass (n ≈ 1.523) to air (n ≈ 1.000), the critical angle is arcsin(1.000 / 1.523) ≈ 41.0°. For flint glass (n ≈ 1.62) it drops to about 38.1°. For diamond (n = 2.417) it is only 24.4°, which is why diamond cutters use pavilion facets at steep angles to maximise internal reflection and brilliance.

What is Fresnel reflectance?

When light crosses an interface it is partly transmitted (refracted) and partly reflected. The Fresnel equations give the reflected fraction for each polarisation. For s-polarisation (electric field perpendicular to the plane of incidence): Rs = ((n₁ cos θ₁ − n₂ cos θ₂) / (n₁ cos θ₁ + n₂ cos θ₂))². For p-polarisation: Rp = ((n₁ cos θ₂ − n₂ cos θ₁) / (n₁ cos θ₂ + n₂ cos θ₁))². For unpolarised light R = (Rs + Rp) / 2. At normal incidence (θ = 0°) this simplifies to R = ((n₁ − n₂) / (n₁ + n₂))². For air-to-glass R ≈ 4 %, which is why uncoated camera lenses lose roughly 4 % per surface.

Why do angles bend toward or away from the normal?

When light enters a denser medium (n₂ > n₁) it slows down and the wavefronts bunch together, bending the ray toward the normal — θ₂ θ₁. You can see this directly in the calculator: entering Water → Crown glass gives a smaller θ₂ than θ₁, while entering Crown glass → Water gives a larger θ₂.

Snell's Law Calculator

Snell’s law is the fundamental equation governing how light (or any wave) bends when it crosses the boundary between two media with different optical densities. Published by René Descartes in 1637 based on the geometric work of Willebrord Snellius, it underpins every lens, fibre-optic cable, camera, microscope, and pair of glasses ever made. This calculator implements the exact formula, lets you solve for any of the four variables, detects total internal reflection, and adds the Fresnel unpolarised reflectance as a bonus.

The formula

n₁ sin θ₁ = n₂ sin θ₂

Symbol	Meaning	Unit
n₁	Refractive index of medium 1 (incident side)	dimensionless
θ₁	Angle of incidence — from the surface normal	degrees
n₂	Refractive index of medium 2 (refracted side)	dimensionless
θ₂	Angle of refraction — from the surface normal	degrees

Because the product n sin θ is conserved across any planar interface, you can rearrange for any single unknown:

θ₂ = arcsin(n₁ sin θ₁ / n₂) — standard forward calculation.
θ₁ = arcsin(n₂ sin θ₂ / n₁) — find the required incident angle.
n₁ = n₂ sin θ₂ / sin θ₁ — identify an unknown medium by measuring both angles.
n₂ = n₁ sin θ₁ / sin θ₂ — find the index of the second medium.

How it works

The calculator applies the exact trigonometric form of Snell’s law — no linear approximations. Angles are converted to radians internally via θ_rad = θ_deg × π / 180 before the sine/arcsine operations. When solving for a refracted angle, it checks whether |n₁ sin θ₁ / n₂| > 1; if so, the refracted ray does not exist and the interface undergoes total internal reflection — the calculator flags this clearly rather than silently returning a NaN.

The critical angle θ_c = arcsin(n₂ / n₁) is displayed whenever n₁ > n₂, reminding you of the TIR threshold for that medium pair.

The Fresnel unpolarised reflectance uses the standard formulae:

R_s = [(n₁ cos θ₁ − n₂ cos θ₂) / (n₁ cos θ₁ + n₂ cos θ₂)]²

R_p = [(n₁ cos θ₂ − n₂ cos θ₁) / (n₁ cos θ₂ + n₂ cos θ₁)]²

R = (R_s + R_p) / 2

This tells you what fraction of the light intensity is reflected — important for anti- reflection coating design and telescope mirror budgets.

Worked example — air into a glass prism

A laser beam in air (n₁ = 1.000293) strikes a crown glass surface (n₂ = 1.523) at 30° from the normal. What is the refracted angle, and how much light is reflected?

Step 1 — Snell’s law:

sin θ₂ = n₁ sin θ₁ / n₂ = 1.000293 × sin(30°) / 1.523
        = 1.000293 × 0.5 / 1.523 ≈ 0.3284
θ₂ = arcsin(0.3284) ≈ 19.17°

The ray bends toward the normal (30° → 19.17°) because glass is denser than air.

Step 2 — Fresnel reflectance:

cos θ₁ = cos 30° ≈ 0.8660,  cos θ₂ = cos 19.17° ≈ 0.9450
Rs = ((1 × 0.8660 − 1.523 × 0.9450) / (1 × 0.8660 + 1.523 × 0.9450))² ≈ 0.0785
Rp = ((1 × 0.9450 − 1.523 × 0.8660) / (1 × 0.9450 + 1.523 × 0.8660))² ≈ 0.0422
R  = (0.0785 + 0.0422) / 2 ≈ 3.5 %

About 3.5 % of the light is reflected — consistent with the ~4 % rule of thumb for uncoated glass at near-normal incidence.

Scenario	n₁	n₂	θ₁	θ₂	R
Air → Water	1.000	1.333	45°	32.1°	2.8 %
Air → Crown glass	1.000	1.523	30°	19.2°	3.5 %
Water → Air	1.333	1.000	30°	41.8°	5.6 %
Water → Air (TIR)	1.333	1.000	50°	—	100 %
Air → Diamond	1.000	2.417	20°	8.2°	17.2 %

Diamond reflects 17 % even at shallow angles — combined with multiple internal TIR reflections, this produces the intense brilliance of a well-cut stone.

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