Reflection and Refraction

• Light is a wave so it does all the same things that other waves do.

• It reflects, refracts, diffracts and can interfere with other light waves.

Reflection

• At any point on a surface we can define a normal.

• The angle between the incoming light ray and the normal is called the angle of incidence.

• The angle between the outgoing light ray and the normal is called the angle of reflection.

• The Law of Reflection is

the angle of incidence is equal to the angle of reflection

• When the light from an object is reflected from a flat reflecting surface (a plane mirror), the reflected light appears to be emitted from an image of the object that is behind the mirror.

• The image of the object is located as far behind the mirror as the real object is in front of it.

• The image of the object is the same size as the object

• Since the light does not actually come from the image in a mirror, the image is called a virtual image.

• If the mirror is curved, these two properties of the image are no longer true.

• If the mirror is convex, the image is smaller and closer to the mirror than the object

• If the mirror is concave, the image is larger and farther away from the mirror than the object.

• If the surface is rough then the normals to the surface point in many different directions.

• If the wavelength of the light is less than the height of the irregularities, the rays of light are reflected in different directions.

• liquids are usually good mirrors because the irregularities of the surface are the height of an atom or two.

Refraction

• The speed of light is different in different materials.

• In addition to the change of speed, the direction the light propagates can be changed.

• Both effects are called refraction.

• Again at the point where the ray meets the boundary between the two materials we can construct a normal

• Consider a ray of light propagating from one material into another material

  • At the point where the ray meets the boundary between the two materials we can construct a normal.

  • The angle of incidence is defined as before

  • The angle of refraction is the angle between the normal the direction of the ray in the new material.

• If the index of refraction in the first material is n1 and the index of refraction if the second is n2, the Law of Refraction is

  • n_1 sin(θ_1)=n_2 sin (θ2)

• If n_2 > n_1 then the angle of refraction must be smaller than the angle of incidence.

• If n_2 < n_1 then the angle of refraction must be greater than the angle of incidence.

• Due to refraction, objects in water appear to be closer to the surface than they really are.

Mirages

• Refraction can occur in a single material if the density changes.

• If the change in density is gradual then the trajectory of the light ray changes smoothly i.e. no sudden jumps.

• For example:

  • as light propagates from denser air into less dense air it curves away from the vertical direction,

  • as light propagates from less dense air into more dense air it curves towards the vertical direction,

• This bending can lead to mirages.

• Light that was traveling towards the ground can be turned around and appear to be coming from the ground.

• If the warm air is above cool air then a different type of mirage is possible.

• Now light that was propagating upwards can be turned around and can appear to be coming from above

• The density of air decreases with height.

• This decrease in density leads to the refraction of light which is most noticeable for sources which are close to the horizon.

• Sources which are actually below the horizon can appear to be above the horizon.

  • the Sun actually drops below the horizon several minutes before it appears to set

Total Internal Reflection

• If light traveling through a material meets a boundary with another material with a lower index of refraction, it is possible that the light does not enter the second material.

• Instead the light is reflected at the boundary, it does not refract.

• This phenomenon is called Total Internal Reflection

• The critical angle is that angle of incidence such that the angle of refraction is 90 degrees.

• The critical angle depends upon the indicii of refraction of the two material.

  • sin(θ_c )= n_2 / n_1

• For air and water the critical angle is 48 degrees

• For glass and air the critical angle is around 43 degrees.

• Using the effect of total internal reflection, a piece of shaped glass acts like mirror.

• Fiber optic cables work on the same principle.

  

• The light cannot escape the fiber because it always meets the glass/air boundary at an angle of incidence greater than the critical angle, no matter the shape of the fiber

Dispersion

• The speed of light in some materials is different for different frequencies of light.

• This means the index of refraction depends upon the frequency which means the angle of refraction depends on frequency.

• This effect is called dispersion.

• Rainbows are a consequence of the dispersive refraction of light by water droplets.

  • Blue/violet light has a higher speed in water than red.

• The different colors emerge from a droplet propagating at different angles.

• An observer on the ground can’t see both rays from a single droplet.

• Instead what an observer sees are rays from different droplets.

• The angle between the red and violet rays which enter the eye is 2 degrees.

• The red arc is on the outside, the violet on the inside

• The arc shape is the edge of a cone whose axis is the line that joins you to the Sun.

  • everyone sees a personal rainbow

• Pilots get to see the entire circle.

• Sometimes a second bow can be seen due to two internal reflections within the droplets.

• The width of this bow is about 3 degrees and the order of the colors is reversed.

• The secondary bow is fainter because less light is reflected twice than the main bow where it is reflected once.

Lenses

• A lens is a shaped piece of a transparent material that disperses or focuses light by means of refraction.

• For visible light the lenses we commonly encounter are made of glass or clear plastic.

  • At other frequencies, lenses can be made from other materials.

• The two basic types of lens are the convex lens which causes light rays to converge, and the concave lens which causes light rays to diverge.

• In both cases the surfaces of the lenses are frequently sections of the surface of a sphere.

• The main use of lenses is to make images:

  • Examples of images are seen on a cinema screen, through telescopes and glasses, and on the retinas of your eyes

  • An image is not what you see on a TV, computer monitor or phone screen, those are sources of light

• Without a lens there would be no image: a lens focuses light so that (ideally) the light at a particular point on the image can only come from one point on the source.

  • An image is a mapping; each point on an image corresponds to a point on the object being viewed.

  • the source may be three dimensional but the image is two-dimensional.

• Images are said to come in two types: real and virtual.

  • a real image is formed at the convergence of light rays

  • a virtual image is formed at the location where divergent light rays appear to diverge from.

  • in practical terms, a real image can be seen on a screen, a virtual image cannot.

• There’s a lot of technical terms associated with lenses

• Principal (Optical) axis: the line joining the centers of curvature of the two lens surfaces

• Focal point: the point at which all the light rays come together

• Focal length: the distance between the center of the lens and either focal point

• Focal plane: the plane passing through the focus and perpendicular to the optical axis

Lens Defects

• No lens can produce a perfect image.

• Lenses have defects that distort the image in some way.

• A distortion of an image is called an aberration.

• There are a number of different aberrations:

  • spherical aberration,

  • chromatic aberration,

  • coma,

  • astigmatism,

• For a spherical lens or mirror, the focus of rays parallel to the optical axis depend upon their distance from the optical axis.

  • this is called spherical aberration

• The different focii will lead to blurring of the image on the chosen focal plane.

  • It is possible to build aspheric lenses that don't suffer this effect but they are much harder (expensive) to manufacture.

• The index of refraction for a material is often different for different colors of light.

• This means the focal length of a lens made from that material will be different for different colors.

• This defect is called chromatic aberration.

• There is no place to put a screen so that all the different colors are in focus at the same time.