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Snell's Law Calculator

Physics

What do you want to find?

Refraction angle θ₂

Active

θ₂ = arcsin((n₁/n₂) · sin θ₁)

The forward problem. Both indices and the incidence angle are known, and you want to know where the ray goes on the far side.

Incidence angle θ₁

θ₁ = arcsin((n₂/n₁) · sin θ₂)

Reverse ray tracing. You know the direction you want the ray to leave in, and you need the entry angle that produces it.

Index of medium 2 (n₂)

n₂ = n₁ · sin θ₁ / sin θ₂

The refractometer experiment: measure both angles across a boundary with a known first medium, and identify the unknown sample.

Index of medium 1 (n₁)

n₁ = n₂ · sin θ₂ / sin θ₁

The same experiment run the other way, when the sample is the medium the light starts in.

Worked examples

The textbook case. The ray bends toward the normal and only about 2 % of the light reflects.

The boundary

So close to vacuum that light slows by only 0.03 %, which is why astronomers still care about it.
The classic textbook medium. Light in water runs at three quarters of c.
Dimensionless, and at least 1. Picking a preset above fills this in.
Dimensionless, and at least 1. Picking a preset above fills this in.

The angles

Measured from the normal, and below 90°.
This is the unknown you are solving for.
Applied to both the inputs and every angle in the results.
Surface readings are converted to the normal before anything is validated.
Drag to watch the refracted ray swing and, past the critical angle, disappear altogether.
Angles are measured from the normal
Every angle here is measured from the normal — the line perpendicular to the surface — not from the surface itself. Reading an angle off the surface instead of the normal is the single most common mistake in refraction problems, and it turns a 40° incidence into a 50° one.

Optional extras

Drives the speed, wavelength and frequency panel. Clear the field to switch that panel off.
Every preset index is quoted near the sodium line unless the medium says otherwise.
Enables the lateral-displacement and apparent-depth panels. Clear the field to switch them off.
0–10.

Refraction angle θ₂

Bends toward the normal
Solved angle (°)
28.839°
Frequency is invariant across the boundary
Frequency does not change when light crosses a boundary. The source sets how many crests leave per second, and crests cannot pile up or vanish at the interface, so the same number must arrive per second on the far side. What changes is the speed — and therefore the spacing between crests, the wavelength. Colour, which follows frequency, is unchanged underwater.
Refractive index is wavelength-specific
Refractive index is wavelength-specific. Water, glass and diamond are dispersive media, so n — and therefore the speed and the wavelength inside them — changes with colour. That is exactly why a prism spreads white light into a rainbow. Each preset above is labelled with the wavelength at which its index was measured; using it far from that wavelength introduces error.
Apparent depth holds only near the normal
The apparent-depth figure holds only for near-normal viewing. Looking steeply across the surface, the shift grows and the image also distorts, which is why a straw looks broken rather than simply shortened.

Ray path

normalθB 53.12°θ₁ = 40.00°reflected 2.45 %θ₂ = 28.84°Medium 1 · n = 1.000Medium 2 · n = 1.333

The denser half-plane is shaded more heavily. Light slows from 2.997e+8 m/s to 2.249e+8 m/s across this boundary, a change of 24.959 %.

Derived quantities

QuantityValueMeaning
Relative index n₂/n₁1.333Above 1 the ray bends toward the normal; below 1 it bends away.
Critical angle θ_cnot definedOnly exists when light leaves the denser medium (n₁ > n₂).
Brewster angle θ_B53.115°Reflected light there is entirely s-polarized.
Deviation θ₁ − θ₂11.161°How far the ray turned at the interface.
Speed of light v₁ / v₂2.997e+8 / 2.249e+8 m/sv = c/n in each medium.
Wavelength λ₁ / λ₂589.127 nm / 442.086 nmCrests crowd together in the slower medium.
Frequency f508.726 THzIdentical on both sides — the reverse check gives 508.726 THz.
Lateral displacement d2.210 mmSideways shift after crossing a parallel slab.
Apparent depth1.333 cmA depth of t in medium 1, viewed from medium 2 near the normal.

How much light gets through

Reflectance R (selected)
2.446 %
R s-polarized
4.308 %
R p-polarized
0.583 %
Transmittance T
97.554 %

At perpendicular incidence this same boundary would reflect only 2.033 %, from R₀ = ((n₁ − n₂)/(n₁ + n₂))².

0%25%50%75%100%0°30°60°90°θB = 53.12°solid R_s · dashed R_p · dotted average
Why the p curve touches zero
At the Brewster angle the reflected and refracted rays are exactly 90° apart and the p-polarized reflectance falls to zero, so everything reflected is s-polarized. Turning a polarizing filter to block that one direction is how glare off water, glass and wet roads is removed.

θ₂ against θ₁

00151530304545606075759090angle of incidence θ₁ (degrees)

With n₁ ≤ n₂ every incidence angle has a refracted ray, and the curve saturates at the maximum refraction angle arcsin(n₁/n₂) rather than reaching 90°.

Speed and wavelength side by side

Speed of light in the medium

Medium 1

2.997e+8 m/s

Medium 2

2.249e+8 m/s

Wavelength in the medium

Medium 1

589.127 nm

Medium 2

442.086 nm

Through a parallel-sided slab

d = 2.210 mmexit ray is parallel to the entry ray

The exit angle equals the entry angle exactly, so the beam direction is unchanged — only its position moves. The optical path through the slab is 2.159 mm longer than the same gap of medium 1.

Apparent depth is a near-normal result
The apparent-depth figure holds only for near-normal viewing. Looking steeply across the surface, the shift grows and the image also distorts, which is why a straw looks broken rather than simply shortened.

Step by step

Snell–Descartes law

n₁ · sin θ₁ = n₂ · sin θ₂

Rearrange for the refraction angle

θ₂ = arcsin((n₁ / n₂) · sin θ₁)

Substitute

sin θ₂ = (1.000293 / 1.333) × sin 40° = 0.48235255

Result

θ₂ = arcsin(0.48235255) = 28.839164°

Brewster angle

θ_B = arctan(n₂ / n₁) = arctan(1.3326095) = 53.115168°

Speeds

v₁ = c/n₁ = 2.997046e+8 m/s; v₂ = c/n₂ = 2.249006e+8 m/s

Fresnel split

R_s = 0.0430809, R_p = 0.00583247, R̄ = 0.0244567, T̄ = 0.975543

Cross-check the law with the solved pair

n₁ sin θ₁ = 0.6429759465; n₂ sin θ₂ = 0.6429759465

Lateral displacement

d = t · sin(θ₁ − θ₂) / cos θ₂ = 0.01 × sin 11.1608° / cos 28.8392° = 0.00220969 m

Every figure is carried at full double precision internally; the decimal places you chose are applied only for display, so no result here is built from a rounded intermediate value.

About This Tool

Snell's Law Calculator – Refraction, Critical Angle and Fresnel Reflection

Light changes direction when it crosses between two transparent media because its phase velocity changes. This Snell's law calculator works that bend out in both directions: give it two refractive indices and one angle and it returns the other angle, or give it two measured angles and it returns the unknown refractive indexand names the material closest to it. Every derived figure an optics problem usually asks for next — the critical angle, the Brewster angle, the speed and wavelength of light on each side, the reflected share of the energy and the sideways shift through a window — is computed from the same unrounded answer.

The law itself

With both angles measured from the normal— the line perpendicular to the surface, never the surface itself — the Snell–Descartes relation is

n₁ · sin θ₁ = n₂ · sin θ₂

Because it is a single equation in four quantities, knowing any three fixes the fourth. Entering air (n = 1.000293) into water (n = 1.333) at 40° gives θ₂ = 28.902°: the ray bends toward the normal because it slowed down. Run it the other way and the ray bends away. The refractive index is just n = c/v, so n = 1.333 means light crawls through water at 2.249 × 10⁸ m/s, three quarters of its vacuum speed.

Total internal reflection and the critical angle

Going from dense to rare, (n₁/n₂) · sin θ₁can exceed 1, and the arcsine has no real solution. That is not an arithmetic failure — it is the physics telling you no refracted ray exists. The cutoff is

θ_c = arcsin(n₂ / n₁)

which is 41.24° for crown glass into air and 24.4° for diamond. Past it the interface becomes a loss-free mirror with no coating at all. Optical fibre, prism binoculars, periscopes, light pipes and the deep sparkle of a well-proportioned brilliant cut are all built on this one effect. Below the cutoff the calculator tells you how many degrees of margin remain.

Frequency does not change at the boundary
The source decides how many wave crests leave per second, and crests cannot pile up or vanish at an interface, so the same number must arrive per second on the far side. What changes is the speed — and therefore the spacing between crests. A 589 nm sodium line becomes 442 nm inside water, yet its colour is unchanged, because colour follows frequency.

How much light actually gets through

Snell's law gives the direction; the Fresnel equations give the share. At normal incidence an air–glass surface reflects about 4 %, which is why an uncoated camera lens with ten elements loses a third of its light. As the angle grows, the s-polarized reflectance climbs steadily, while the p-polarized component drops to exactly zero at the Brewster angle θ_B = arctan(n₂/n₁) — about 56° for glass. Everything reflected there is s-polarized, which is exactly what a polarizing filter is rotated to remove when you shoot through a shop window or across a lake.

Slabs, apparent depth and everyday refraction

A ray crossing a parallel-sided slab exits parallel to the way it came in, but displaced sideways by d = t · sin(θ₁ − θ₂) / cos θ₂. Ten millimetres of BK7 at 30° shifts a laser beam by 3.41 mm— enough to matter in an alignment budget. The same geometry explains why a pool looks shallower than it is: viewed near the vertical, an object appears at n₂/n₁ of its real depth, three quarters for water. It is also why a straw looks broken at the waterline and why the sun stays visible for a couple of minutes after it has geometrically set.

Index depends on wavelength
Every preset index here is quoted at the wavelength shown beside it, usually the 589.3 nm sodium line. Real media are dispersive: crown glass varies by roughly 0.5 % across the visible spectrum, which is precisely why a prism spreads white light into a rainbow. Using one index for all colours is an approximation, and a poor one near a material's absorption edges.

The multi-layer mode chains up to ten media in sequence and traces the ray through every boundary, flagging the first one that reflects it totally. Because n · sin θis conserved across the whole stack, the exit angle depends only on the first and last media, never on what sits between them — a result worth checking against your own working, and one the step-by-step panel makes explicit.

Frequently Asked Questions

Is the Snell's Law Calculator free?

Yes, Snell's Law Calculator is totally free :)

Can I use the Snell's Law Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Snell's Law Calculator?

Yes, any data related to Snell's Law Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this Snell's law calculator work?

Pick which of the four quantities in n₁·sin θ₁ = n₂·sin θ₂ is unknown, choose the two media from the preset library or type indices directly, and enter the angles you know. The tool solves the rearranged form for your unknown, then computes every derived figure from the unrounded answer: the critical angle, the Brewster angle, the speed and wavelength of light in each medium, the Fresnel reflectance split, the deviation, and the lateral shift through a slab. All angle arithmetic happens in radians internally, with degrees, radians or gradians converted only at the input and output boundaries.

Why must angles be measured from the normal and not the surface?

Because Snell's law is written for the normal — the line perpendicular to the interface. Reading 40° off the surface instead of the normal means you actually have 50° incidence, and the answer changes by several degrees. If your protractor is against the glass, switch the "measure angle from" control to Surface and the tool converts the reading for you before validating it, so the value the physics sees is always the one from the normal.

What is the critical angle, and when does the calculator report total internal reflection?

The critical angle θ_c = arcsin(n₂/n₁) exists only when light travels from a denser medium into a rarer one, that is when n₁ > n₂. Beyond it, (n₁/n₂)·sin θ₁ exceeds 1, the arcsine has no real solution and no light crosses the boundary at all. Instead of returning NaN, the calculator reports the total-internal-reflection state explicitly, shows how far past the cutoff you are, draws the reflected ray at the same angle as the incident one, and sets the reflectance to exactly 1.

What is the Brewster angle used for?

At θ_B = arctan(n₂/n₁) the p-polarized reflectance falls to zero, so every photon reflected from the surface is s-polarized and the reflected and refracted rays sit exactly 90° apart. That single fact is the basis of polarizing filters: rotating one to block the s direction removes glare from water, wet roads and shop windows almost completely. For an air-to-glass boundary the angle is about 56°, which is why the effect is strongest on low-angle sunlight.

How accurate are the built-in refractive indices?

Each preset is quoted at the wavelength shown next to it, usually the 589.3 nm sodium line or the 1550 nm telecom window, and is good to the digits displayed for a typical sample. Refractive index is genuinely wavelength-dependent, though — that dispersion is why a prism makes a rainbow — so using a sodium-line index for blue light introduces a real error, around 0.5 % for crown glass across the visible range. Temperature, pressure and composition shift the figure too, which matters most for gases and liquids.

Why does the reflectance rise as the angle increases?

The Fresnel equations make the reflected share depend on both indices and on the angle. At normal incidence an air-glass surface reflects only about 4 %, so the transmitted beam loses almost nothing. As the angle grows the s-polarized reflectance rises steadily while the p-polarized component first falls to zero at the Brewster angle and then rises, and near grazing incidence both approach 100 % — which is why a still lake looks like a mirror when you view it from a low angle but transparent when you look straight down.