Refraction and Snell's law

Why light bends

Refraction occurs when light crosses from one medium into another and changes speed. If the ray enters at an angle, the change in speed changes the direction of propagation. This bending is responsible for lenses, prisms, mirages, and the apparent bending of objects partly submerged in water.

The speed of light in vacuum is

$capprox 3.00 imes 10^8 m/s$

In a material, light travels more slowly. The index of refraction is

n=rac{c}{v}

where $v$ is the speed of light in the material. Since $v<c$ for most transparent materials, $n>1$.

Snell's law

When light passes from medium 1 to medium 2, the angles satisfy Snell's law:

$n_1sin heta_1=n_2sin heta_2$

Angles are measured from the normal to the surface. If light enters a medium with larger refractive index, it bends toward the normal. If it enters a medium with smaller refractive index, it bends away from the normal.

For example, light passing from air into glass bends toward the normal because glass has a larger refractive index than air.

Frequency and wavelength

When light crosses a boundary, its frequency remains the same. The source determines frequency, and the electric field oscillations must match at the boundary. But speed changes, so wavelength changes.

Using

$v=flambda$

and

v=rac{c}{n}

the wavelength in a material is

lambda=rac{lambda_0}{n}

where $lambda_0$ is the wavelength in vacuum.

Deriving the direction change

Snell's law can be understood through wavefronts. When one side of a wavefront enters a slower medium first, that side slows while the other side continues faster, causing the wavefront to rotate. The ray direction, perpendicular to the wavefront, changes.

Snell's law can also be derived from Fermat's principle: light takes a path of stationary travel time. The fastest path through two media is not generally the shortest geometric distance.

Apparent depth

Refraction makes underwater objects appear closer to the surface than they really are. Rays from the object bend away from the normal as they leave water into air. Your eye traces the rays backward in straight lines, placing the object at a shallower virtual location.

For near-normal viewing, apparent depth is approximately

d_{app}approx rac{d}{n}

for an object in a medium of refractive index $n$ viewed from air.

Dispersion

Refractive index often depends on wavelength. This is called dispersion. In glass, shorter wavelengths of visible light are usually refracted more strongly than longer wavelengths. A prism separates white light into colors because each wavelength bends by a different amount.

Dispersion also causes chromatic aberration in lenses, where different colors focus at slightly different positions.

Refraction and energy

Refraction changes direction and wavelength, but frequency stays fixed. Some light may also reflect at the boundary. The amount reflected and transmitted depends on angle, polarization, and refractive indices. Full treatment requires electromagnetic boundary conditions, but ray optics captures the basic geometry.

The big idea

Refraction is caused by a change in light speed at a boundary. Snell's law predicts how rays bend, while refractive index describes how much a material slows light. Frequency remains constant across a boundary, wavelength changes, and dispersion makes refractive index depend on color.