Refraction Using Huygens' Principle

Start with a plane wavefront incident on a plane boundary. Mark the point that reaches the second medium first, draw its secondary wavelet of radius v2t in the second medium, and mark the distance v1t travelled by the other end in the first medium during the same time. Draw the refracted wavefront as the tangent to the secondary wavelet. Use the two right triangles to obtain sin theta1 / sin theta2 = v1 / v2, then substitute n = c/v to get n1 sin theta1 = n2 sin theta2.

Draw a plane boundary, normal at the point of incidence, incident wavefront, refracted wavefront, incident ray and refracted ray. Label theta1 and theta2 with respect to the normal, and show different wave speeds v1 and v2.

A common wrong answer is n2 sin theta1 = n1 sin theta2, caused by interchanging media. Keep the refractive index with the sine of the angle in the same medium.

Refraction by Huygens' principle explains the bending of a plane wavefront at a boundary because light travels with different speeds in two media, leading to the relation n1 sin theta1 = n2 sin theta2.

Refraction by Huygens' principle explains the bending of a plane wavefront at a boundary because light travels with different speeds in two media, leading to the relation n1 sin theta1 = n2 sin theta2. Assumptions: the incident wavefront is plane, the boundary is plane, and light speeds are v1 and v2 in the two media. Geometry gives sin theta1 / sin theta2 = v1 / v2. Using n = c/v, this becomes n1 sin theta1 = n2 sin theta2. Here n1 and n2 have no SI unit, while v1 and v2 are in m s^-1 and theta1, theta2 are measured from the normal. When light enters glass from air, its speed decreases. The refracted ray bends towards the normal, and the refracted wavefront changes orientation accordingly.

When one end of an incident wavefront reaches the second medium earlier than the other end, that part begins to travel with the speed of the second medium while the remaining part is still in the first medium. The change in speed changes the direction of the refracted wavefront. Rays drawn normal to the incident and refracted wavefronts give the incident and refracted ray directions. Assumptions: the incident wavefront is plane, the boundary is plane, and light speeds are v1 and v2 in the two media. Geometry gives sin theta1 / sin theta2 = v1 / v2. Using n = c/v, this becomes n1 sin theta1 = n2 sin theta2. Here n1 and n2 have no SI unit, while v1 and v2 are in m s^-1 and theta1, theta2 are measured from the normal. Light travels from air, n1 = 1.00, into glass, n2 = 1.50, at theta1 = 30 degree. Using n1 sin theta1 = n2 sin theta2: 1.00 x sin 30 degree = 1.50 x sin theta2. So sin theta2 = 0.50/1.50 = 0.333 and theta2 = 19.5 degree. If c = 3.0 x 10^8 m s^-1, the speed in glass is v2 = c/n2 = 2.0 x 10^8 m s^-1, confirming that the ray bends towards the normal because light slows down. It appears as a derivation of Snell's law from Huygens' principle, a ray-wavefront diagram, or a numerical using refractive indices and angles. A common wrong answer is n2 sin theta1 = n1 sin theta2, caused by interchanging media. Keep the refractive index with the sine of the angle in the same medium.