Refraction is the change of direction of a light wave caused by a change in speed as the wave crosses a boundary between materials.

The index of refraction of a material is the speed of light in a vacuum divided by the speed of light in the material, n = c/v.

The index of refraction is greater than one for any material other than a vacuum.

Snell’s law of refraction - When light travels from a material with refractive index n 1 into a material with refractive index n 2, the refracted ray, incident ray, and normal all lie in the same plane. The angle of refraction q 2 is related to the angle of incidence q 1 by n 1 sin  1 = n 2 sin  2.

Ex. 1 - A light ray strikes an air/water surface at an angle of 46° with respect to the normal. The refractive index for water is Find the angle of refraction when the direction of the ray is (a) from air to water and (b) from water to air.

When a ray of light passes obliquely from a medium of lower index of refraction to one of higher index of refraction, it is bent toward the normal to the surface. A ray of light passing from a medium of higher index of refraction to one of lower index of refraction is bent away from the normal to the surface.

When light energy is simultaneously reflected and refracted at a boundary, the total energy must remain constant. When light is directed along the normal, most all is refracted and little is reflected.

But when the angle of incidence is nearly 90°, most of the energy is reflected and little is refracted.

Ex. 2 - A searchlight on a yacht is used at night to illuminate a sunken chest. The chest in water 3.3 m deep, and the incident ray strikes the water at a point 2.0 m from a point directly above the chest. At what angle of incidence q 1 should the light be aimed?

The chest (or any object) is seen as being at a depth less than the actual depth. This virtual image is at an apparent depth that can be found using this formula: d’ = d(n 2 /n 1 ). d’ is the apparent depth d is the actual depth n 2 is the medium of the observer n 1 is the medium of the object

Ex. 3 - A swimmer is treading water at the surface of a 3.00-m-deep pool. She sees a coin on the bottom directly below. How deep does the coin appear to be?

Ex. 4 - A swimmer is under water and looking up at the surface. Someone holds a coin in the air, directly above the swimmer’s eyes. To the swimmer, the coin appears to be at a certain height above the water. Is the apparent height of the coin greater than, less than, or the same as its actual height?

A transparent slab of material displaces a ray of light. The amount of the displacement depends on the angle of incidence, the thickness of the slab, and its refractive index. The ray that enters and the ray that exits are parallel if the surfaces of the slab are parallel.

When a light wave enters a material of higher index of refraction, the speed decreases, the wavelength decreases, but the frequency remains unchanged. The frequency of a light wave is constant.

When light passes from a medium of larger index of refraction to a medium of lower index of refraction, the refracted ray bends away from the normal.

The angle of incidence is smaller than the angle of refraction. As the angle of incidence increases, the angle of refraction does also; therefore, the angle of refraction reaches 90° before the angle of incidence.

At an incident angle called the critical angle  C, the angle of refraction is 90°.

At an angle of incidence above the critical angle all the incident light is reflected at the boundary back into the medium; this is total internal reflection.

Total internal reflection can only occur when light moves from a higher- index medium to a lower-index medium.

Using Snell’s law, this formula for the critical angle can be obtained: sin  C = n 2 sin 90°/n 1. Since sin 90° = 1, sin  C = n 2 /n 1 if n 1 > n 2.

Ex. 5 - A beam of light is propagating through diamond (n 1 = 2.42) and strikes a diamond- air interface at an angle of incidence of 28°. (a) Will part of the beam enter the air (n 2 = 1.00) or will the beam be totally reflected at the interface? (b) Repeat part (a), assuming that the diamond is surrounded by water (n 2 = 1.33).

Optical instruments like binoculars and telescopes use prisms and total internal reflection to redirect rays of light. Mirrors reflect a percentage of light; total internal reflection is more efficient because it is 100% reflection.

Total internal reflection is also used in optical fiber to keep the light rays within the fiber.

A prism bends light as it enters and as it leaves the prism. Its triangular shape causes the light to be refracted twice in the same direction.

The index of refraction is different for different colors. Different wavelengths are refracted different amounts, so the light is separated into a spectrum of colors; this is called dispersion.

Converging lens - convex lens, thicker in the middle than at the edges. Diverging lens - concave lens, thicker at the edges than in the middle.

The principal axis passes through the two centers of curvature of the two surfaces of the lens.

Rays that are parallel to the PA converge at the principal focus of the lens. If they actually pass through this point it is a real focus.

If rays do not pass through a principal focus, it is a virtual focus.

The positions of the foci on the principal axis depend on the index of refraction of the lens.

The focal length of a lens is the distance between the optical center of the lens and the principal focus.

Parallel rays that are not parallel to the PA are focused on the focal plane.

Convex lenses form images like concave mirrors, concave lenses form images like convex mirrors.

Images formed by converging lenses: Case #1 Object at an infinite distance.

Case #2 Object at a finite distance beyond the twice the focal length

Case #3 Object at a distance equal to twice the focal length

Case #4 Object between one and two focal lengths

Case #5 Object at principle focus

Case #6 Object at a distance less than one focal length away

Diverging lenses