Snell's Law and Total Internal Reflection Explained (With the Critical Angle)

Why light bends

Light travels more slowly in glass or water than in air. The refractive index n says how much: light in a material of index n moves at c ÷ n, so in water (n = 1.333) it covers about 225,000 km each second instead of 300,000. When a beam crosses a boundary at an angle, one edge of it reaches the slower material first and slows first, and the beam swings round, like a column of marchers wheeling as one end steps onto mud. That turning is refraction.

Snell's law

n₁ sin θ₁ = n₂ sin θ₂

θ₁ is the angle of incidence and θ₂ the angle of refraction, both measured from the normal, the line square to the surface, not from the surface itself.
Going into a denser material (n₂ > n₁) the ray bends toward the normal; coming out of one it bends away.
A ray that hits the surface straight on (θ₁ = 0) goes straight through, only slower.

A worked example

A laser in air hits a block of crown glass (n = 1.517) at 40° from the normal.

sin θ₂ = (1.000 ÷ 1.517) × sin 40° = 0.659 × 0.643 = 0.424.
θ₂ = arcsin 0.424 = 25.1°.

Not all the light gets in. The Fresnel equations give the reflected share; at 40° from air into this glass it is about 4.8%, and straight on it is ((n − 1) ÷ (n + 1))² = 4.2%. That is why a sheet of window glass, with two surfaces, reflects about 8% and shows you a faint reflection at night.

The critical angle and total internal reflection

Going from a denser material into a less dense one, the refracted ray bends away from the normal, and at a large enough angle it would have to leave at more than 90°. It cannot, so no light gets out at all and every bit of it reflects. The angle where this starts is the critical angle:

sin θ_c = n₂ ÷ n₁ (only when n₁ > n₂)

Water to air: θ_c = arcsin(1 ÷ 1.333) = 48.6°. Look up at the surface from under water and beyond that angle it is a mirror.
Crown glass to air: 41.2°. A 45° prism therefore reflects light completely, better than a silvered mirror, which is how binoculars and periscopes fold their light paths.
Diamond to air: 24.4°. Most light entering a well-cut diamond is trapped and bounced around inside before it leaves through the top, which is where the sparkle comes from.

Even before the critical angle the reflected share climbs steeply: from glass into air it is 4% straight on, about 10% at 35° and 100% at 41.2°.

Brewster's angle

At tan θ_B = n₂ ÷ n₁ (56.6° from air into crown glass, 53.1° onto water) light polarized in the plane of incidence is not reflected at all, so the reflected light is completely polarized parallel to the surface. Polarizing sunglasses block that direction, which is why they cut the glare off water and roads.

Prisms and dispersion

The refractive index depends slightly on the wavelength. For crown glass it is about 1.531 for violet light at 400 nm and 1.513 for red at 700 nm, so violet bends more at each face of a prism and white light fans out into a spectrum. Cauchy's formula, n(λ) = A + B ÷ λ², describes this well across the visible range, and it is what the simulation uses. The deviation of a beam through a prism is smallest when it crosses symmetrically, and that minimum gives the index: n = sin((A + δ_min) ÷ 2) ÷ sin(A ÷ 2), with A the apex angle. For a 60° crown glass prism δ_min is about 38.6°.

Optical fibers

A fiber traps light by total internal reflection: a glass core with a slightly lower-index cladding around it. Light that enters the end within the acceptance angle hits the side beyond the critical angle every time and is carried for kilometers. The numerical aperture is NA = √(n_core² − n_clad²) = sin θ_max (in air). A core of 1.47 in a cladding of 1.455 gives NA = 0.21 and an acceptance angle of about 12°.

Using the simulation

In the Refraction Simulator, pick the two materials and set the angle. The beam, its faint reflection and the normal are drawn, and the dots running along the beam slow down by the factor n in each material. The chart plots the reflected share against the angle for both polarizations, with the current angle, Brewster's angle and any critical angle marked. Try these:

Water into air, then sweep the angle: watch the refracted beam lie down along the surface and disappear at 48.6°.
The prism with white light: read the minimum deviation, then change flint glass for crown glass and see the spectrum widen.
The fiber at 30° and then at 80° with water around the glass: past the acceptance angle the light leaks out of the side.

What the model assumes

Rays. Light is treated as thin rays that travel in straight lines. That is geometric optics, accurate when everything the light meets is much larger than its wavelength (about half a micrometer).
Flat, clean, sharp boundaries between two uniform, transparent materials, with no absorption or scattering inside them. The light that does not reflect is all carried on.
Unpolarized light for the reflected share, the average of the two polarizations; the chart shows each separately.
Indices from Cauchy's formula, fitted to typical values for each material at 20 °C. Real glasses vary by type and batch in the third decimal place.
Only the main beam is followed through a prism or fiber, plus the first reflection. Each surface really sends back a few percent more, which makes faint ghost beams the simulation does not draw.
The fiber is the core in the material you choose as the first one, standing in for the cladding, so with air around it the acceptance angle is 90°. Choose water around glass to see a realistic, limited acceptance angle.

Edge cases

θ₁ = 0: the beam goes straight through with no bending, and the reflected share is at its minimum for that pair of materials.
Equal indices: nothing happens at the boundary, no bending and no reflection. A glass rod disappears in oil of the same index, a trick used to check a glass's index.
Exactly at the critical angle the refracted ray runs along the surface and carries no energy; a hair past it, the reflection is total.
Close to 90° (grazing incidence) almost all the light reflects from any surface, even from air into glass, which is why a road or a lake looks like a mirror when you look along it.
Prism at a steep apex angle or a high index: the beam meets the second face beyond the critical angle and is reflected back into the prism instead of leaving, and there is no minimum deviation.

Where the model stops being right

Small things. Slits, gratings, droplets and fibers only a few wavelengths across diffract light, which rays cannot describe; the Double-Slit Simulator shows the wave picture. Real single-mode telecom fibers have cores of about 9 µm and must be treated as waveguides.
Frustrated total internal reflection. Even in total internal reflection, a weak evanescent wave reaches about a wavelength beyond the surface. Bring another piece of glass that close and light tunnels across, which ray optics says cannot happen.
Absorbing and metallic materials have a complex refractive index, and the simple Fresnel formulas used here do not apply to them. Coatings a fraction of a wavelength thick, such as anti-reflection layers, work by interference, which the simulation does not include.
Wavelengths outside the visible. Cauchy's formula fails near absorption bands, in the ultraviolet and infrared, where the index changes sharply; Sellmeier's equation is used there instead.
Uneven materials. Air of varying temperature, as over a hot road, has an index that changes gradually, so light curves instead of bending at one surface, producing mirages. The simulation has only sharp boundaries.
Crystals such as calcite are birefringent: their index depends on the polarization and direction, and one ray splits into two.

Related tools

The Lens and Mirror Simulator uses refraction at curved surfaces to form images, with the thin lens equation worked through.

Snell's Law, Refraction and Total Internal Reflection