Snell's Law Calculator

The Snell's Law Calculator solves any unknown in the equation \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \) — angle of refraction, angle of incidence, either refractive index, or the critical angle for total internal reflection. Pick from a library of common materials (water, crown glass, diamond, optical-fiber core and cladding, sapphire, and more) or enter your own indices, watch an interactive light-ray diagram with animated photons, and see the speed and wavelength of light inside each medium.

How to Use This Snell's Law Calculator

1. Choose what you want to solve for: the angle of refraction θ₂, the angle of incidence θ₁, the refractive index n₁ or n₂, or the critical angle for total internal reflection.
2. Pick the two media. Use the dropdowns to choose from common materials, or select "Custom" and type your own refractive index.
3. Fill in the angles you know. The angle field for the variable you are solving for is greyed out automatically.
4. Optional — enter a vacuum wavelength in nanometres (589 nm is the textbook sodium-D yellow line) to also see the wavelength shrink inside each medium.
5. Press Calculate and read the result, the step-by-step derivation, an animated ray diagram, and bonus outputs like Brewster's polarization angle and the speed of light in each medium.

What Makes This Calculator Different

The Snell's Law Equation

When light crosses the boundary between two transparent media, the angles (measured from the normal — the perpendicular to the boundary) are related by:

\[ n_1 \sin\theta_1 \;=\; n_2 \sin\theta_2 \]

where \(n_1\) and \(n_2\) are the refractive indices of medium 1 and medium 2, and \(\theta_1\) and \(\theta_2\) are the angle of incidence and the angle of refraction respectively. The refractive index of a medium is defined as the ratio of the speed of light in vacuum to the speed of light in that medium, \(n = c / v\), so a higher index always means light travels more slowly.

Critical Angle and Total Internal Reflection

When light tries to go from a denser medium to a less dense one (n₁ > n₂), the refracted ray bends away from the normal. As θ₁ grows, θ₂ approaches 90° — meaning the refracted ray would skim along the boundary. At the special angle

\[ \theta_c \;=\; \arcsin\!\left(\dfrac{n_2}{n_1}\right) \]

and above it, no real refracted ray exists — all the light bounces back into medium 1. This is total internal reflection, and it is the optical principle behind fiber optic cables, prisms in binoculars, and the way diamonds throw back so much light.

Brewster's Angle (Bonus Output)

Brewster's angle is the angle of incidence at which the light reflected from a transparent surface is completely polarized in the direction perpendicular to the plane of incidence:

\[ \theta_B \;=\; \arctan\!\left(\dfrac{n_2}{n_1}\right) \]

Polarized sunglasses use this fact: the glare reflected off water, roads, and snow near Brewster's angle is mostly polarized horizontally, and a vertical polarizer in the sunglasses blocks most of it. Photographers use a circular polarizing filter for the same reason — to cut reflections from glass and water.

Refractive Indices of Common Materials (at 589 nm)

Worked Example: A Coin in a Pool

Light from a coin at the bottom of a swimming pool travels up through water (n₁ = 1.333) and exits into air (n₂ = 1.0003). If the light leaves the coin at 40° from vertical (the normal), the angle at which it emerges into the air is

\[ \theta_2 \;=\; \arcsin\!\left(\dfrac{1.333}{1.0003} \sin 40°\right) \;\approx\; 59.0° \]

The ray bends away from the normal (because it is going from a denser to a less dense medium), which is exactly why the coin looks shallower and offset from where it actually is. Push the angle higher and at θ₁ ≈ 48.6° the calculator switches to total internal reflection — no light escapes the water at that grazing angle, which is why you cannot see out of a pool sideways from underwater.

Worked Example: Fiber Optic Cable

A typical step-index optical fiber has a core with n₁ ≈ 1.475 and cladding with n₂ ≈ 1.460. The critical angle is

\[ \theta_c \;=\; \arcsin\!\left(\dfrac{1.460}{1.475}\right) \;\approx\; 81.8° \]

Any ray that bounces inside the core at greater than 81.8° from the normal is totally reflected at every wall, so light injected into the end of the fiber stays trapped along the length and can travel kilometers before significant loss. That is the entire physical basis of modern long-distance internet.

Why Light Bends — A Wavefront Intuition

Picture a wavefront of light arriving at the boundary at an angle. The first edge of the wavefront to enter the new medium slows down (or speeds up, if it is going into a lower-index medium) before the rest of the wavefront does. That mismatch in speed across the wavefront twists the direction of the wave, just like a marching band pivots when the line crosses from pavement to mud. Snell's Law is exactly the geometry of that pivot.

Speed of Light and Wavelength in a Medium

Because \(n = c/v\), the speed of light in a medium is \(v = c/n\). In water (n = 1.333) the speed is about 225,000 km/s, in crown glass about 197,500 km/s, and in diamond only 124,000 km/s. The frequency of the light is the same on both sides of the boundary (it has to be — the boundary cannot create or destroy oscillations), so the wavelength inside the medium is

\[ \lambda_{\text{medium}} \;=\; \dfrac{\lambda_{\text{vacuum}}}{n} \]

This is why 589 nm sodium-yellow light has a wavelength of only about 442 nm inside water, even though your eye still sees the same yellow color.

Frequently Asked Questions

What is Snell's Law in simple terms?
When light goes from one transparent material into another at an angle, it bends. Snell's Law is the exact recipe: the refractive index times the sine of the angle (from the normal) is the same on both sides — n₁ sin θ₁ = n₂ sin θ₂.

What is the critical angle?
When light goes from a denser to a less dense medium, there is a steepest angle of incidence beyond which no refracted ray exists — all the light is reflected back. That angle is the critical angle, given by arcsin(n₂/n₁). It is the mechanism behind fiber optics.

What is Brewster's angle?
It is the angle of incidence at which the reflected light is fully polarized perpendicular to the plane of incidence: θ_B = arctan(n₂/n₁). Polarized sunglasses and photo polarizers work because reflections from water, glass, and roads near this angle are strongly polarized.

Why does light bend when it enters water?
Light travels more slowly in water than in air. When a wavefront arrives at an angle, one edge of the front slows down before the rest does, twisting the wave direction toward the normal. Snell's Law fixes the exact amount of twist.

Does the wavelength of light change in a medium?
Yes. Frequency stays the same when light crosses a boundary, but wavelength shortens by a factor of n: λ_medium = λ_vacuum / n. The color you see is unchanged because color is set by frequency, not by wavelength.

Can the refractive index be less than 1?
For visible light in ordinary materials, no — n is always ≥ 1, with vacuum exactly equal to 1. Engineered metamaterials and certain regimes (X-rays in matter, plasmas) can have phase indices below 1 or even negative, but this calculator covers the standard visible/optical regime.

Why do diamonds sparkle?
Diamond has a very high refractive index (n ≈ 2.417), which gives a small critical angle of about 24.4°. Most light that enters a well-cut diamond hits the back facets above that angle, gets totally internally reflected, bounces around inside, and exits through the top — producing the characteristic "fire."