Critical angle of sapphire into air, with Snell's law shown

Light traveling inside sapphire can only get out into air if it meets the surface close enough to the normal. Past the critical angle none of it escapes and the surface works as a perfect mirror from the inside. The calculator below is set to sapphire as the first medium and air as the second, with the angle of incidence left empty, so it shows the critical angle on its own.

The number comes from one line of Snell's law: at the critical angle the refracted ray skims along the surface, so its sine is 1 and the sine of the critical angle equals the index of air divided by the index of sapphire. The higher the index, the smaller the angle, which is why diamond keeps light bouncing inside while water lets most of it out.

Type an angle of incidence to follow a single ray instead, or add a depth to see how deep an object in sapphire looks from above.

Problem: a pool that looks shallower and a window that turns into a mirror

The deep end marked 8 ft looks closer to 6 ft from the edge. A window that was clear all afternoon shows your reflection after dark. A diver looking up sees the sky squeezed into a circle. All three come from light changing speed at a surface, bending, and partly bouncing back.

Pick where the light starts and where it goes, then type the angle it arrives at, or leave the angle empty for the critical angle alone. The result gives the angle it leaves at, the critical angle, the share reflected, the Brewster angle and the speed of light on both sides. Air into water at 45° comes out at 32.05°, with 2.78 % reflected.

Solution: Snell's law and the reflection that comes with it

The bend. n₁ sin θ₁ = n₂ sin θ₂, written out with your numbers.

The point where bending stops. Past the critical angle the equation would need a sine above 1, and the result says so instead of printing an angle that does not exist. With no angle typed, it gives that critical angle on its own.

The share that bounces. Fresnel reflectance for ordinary light, which decides whether a surface looks clear or like a mirror.

Decision table: what light does at a water surface

Water has an index of 1.333 and air 1.0003. The same surface behaves very differently depending on which side the light comes from and how steeply it arrives.

Light travels	Arrives at	Leaves at	Reflected	What you see
Air into water	0°	0.00°	2.03 %	looking straight down, clear water
Air into water	45°	32.05°	2.78 %	a straw looks broken at the surface
Air into water	80°	47.65°	34.78 %	low sun, glare off a lake
Water into air	40°	58.93°	5.54 %	a diver sees the shore, stretched
Water into air	48.63°	90.00°	100.00 %	the critical angle, the edge of the circle
Water into air	60°	none	100.00 %	the underside of the surface is a mirror

Four places the same law shows up

How deep is the deep end? Water to air, straight down, real depth 8 ft: it appears 6.00 ft deep, 75.0 % of the truth. At a slant from the side it looks shallower still.

Why the window becomes a mirror at night. Air into crown glass head-on reflects 4.21 % at each of the pane's two surfaces. In daylight the view outside drowns that out; after dark the reflected room is the brightest thing in the frame.

Why a diamond sparkles more than glass. Diamond into air has a critical angle of 24.45°, crown glass 41.26°. Light inside a diamond escapes through a narrow cone, so it bounces between facets first, and head-on the surface reflects 17.19 %.

The circle of sky above a diver. Everything above the water reaches the eye inside a cone twice the critical angle wide, about 97.3°. Outside it, at 60° for example, the surface reflects 100 % of the light from below.

From the first medium to the angle, in the order of the form

Medium 1 - where the light comes from. Thirteen media are listed; "custom" takes your own index.
Medium 2 - where it goes. For a diver looking up, that is air.
Custom index - only if you picked custom: a number from 1 (vacuum) up to 5. Diamond is 2.417.
Angle of incidence - measured from the normal, the line perpendicular to the surface: 0° is straight in, 90° skims the surface. Leave it empty to see only the critical angle.
Real depth (optional) - how deep something sits in medium 1, in any unit; the apparent depth comes back in the same unit.

Cheat sheet: index, speed and critical angle

Indices at the sodium line, 589 nm. The critical angle is for light leaving into air; the last column is the share reflected head-on from air.

Medium	n	Light speed (km/s)	Critical angle	Reflected head-on
Ice	1.31	228,849	49.78°	1.80 %
Water	1.333	224,901	48.63°	2.03 %
Ethanol	1.361	220,274	47.31°	2.33 %
Fused silica	1.458	205,619	43.32°	3.47 %
Glycerol	1.473	203,525	42.77°	3.65 %
Acrylic (PMMA)	1.49	201,203	42.17°	3.87 %
Crown glass (N-BK7)	1.5168	197,648	41.26°	4.21 %
Flint glass	1.62	185,057	38.13°	5.59 %
Sapphire	1.77	169,374	34.41°	7.72 %
Cubic zirconia	2.16	138,793	27.59°	13.47 %
Diamond	2.417	124,035	24.45°	17.19 %

Questions readers ask about bending light

Does light bend toward or away from the normal?

Toward it when entering a medium with a higher index, where light slows down: air into water at 45° leaves at 32.05°. Away from it when entering a lower index: water into air at 40° leaves at 58.93°.

Why is there no critical angle going from air into water?

Because the refracted ray always bends toward the normal, so it can never reach 90°. Even light skimming the surface at 90° enters, at 48.63°. Total internal reflection only happens on the way out. For the critical angle of a material, pick it as the first medium, air as the second and leave the angle empty: water gives 48.63°, diamond 24.45°.

How does a fiber optic cable keep light inside?

The core has a slightly higher index than the cladding around it, so light running along it meets the boundary past the critical angle and is reflected every time. Flint glass in air would give 38.13°; real fibers use a much smaller index step and a larger critical angle.

Is a higher refractive index the same as a denser material?

Not in mass per volume. "Optically denser" only means a higher index: acrylic, far lighter than glass, is close to glass optically.

Why does the reflected share jump near 90 degrees?

Fresnel reflectance climbs slowly and then steeply as the angle approaches grazing. Air into water reflects 2.03 % head-on, 2.78 % at 45°, 34.78 % at 80° and all of it at 90°, which is why a lake at sunset reflects the sky like a sheet of metal.

Do the indices change with color and temperature?

Slightly. Blue light sees a higher index than red, which is how a prism splits white light. The table uses the standard sodium line at 589 nm; for another wavelength, choose custom and type its index.

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What is the critical angle of sapphire going into air?

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