Snell's law
/ SNEL /
Refraction tells you that light bends when it changes medium, but by how much? Bend a little or bend a lot? Snell's law is the exact rulebook. It is the equation that takes the angle a ray comes in at and tells you the precise angle it leaves at. It answers the sharp question refraction leaves open: given the two materials and the incoming angle, what is the outgoing angle?
Measure both angles from the normal (the perpendicular to the surface). Let theta1 be the angle in medium 1, which has index of refraction n1, and theta2 the angle in medium 2, with index n2. Snell's law says n1 sin(theta1) = n2 sin(theta2). The product of the index and the sine of the angle is the same on both sides of the boundary. So going into a denser (higher-index) medium, sin(theta2) must shrink, meaning theta2 is smaller and the ray bends toward the normal, exactly what makes a pool look shallow. Going the other way, into a lower-index medium, the ray bends away from the normal.
Snell's law is the master equation of geometric optics: every lens, prism, camera, and pair of glasses is designed with it. It is named after the Dutch astronomer Willebrord Snellius, though the same relation was found independently by Ibn Sahl centuries earlier and by Descartes. An honest caveat: when light tries to leave a denser medium for a thinner one at a steep enough angle, Snell's law would demand sin(theta2) greater than 1, which is impossible, no refracted ray can exist. At that point the light is entirely reflected back, the phenomenon of total internal reflection.
A ray hits a water surface (n1 = 1.00 for air) at theta1 = 40 degrees from the normal. Then sin(theta2) = (1.00/1.33) sin(40 deg) = 0.483, so theta2 = 29 degrees. The ray bends toward the normal as it enters the slower water.
n1 sin(theta1) = n2 sin(theta2); angles measured from the normal.
Both angles are taken from the normal, never from the surface. And Snell's law works both ways along the same path, light retraces its route if you send it back.