Electromagnetic waves with wavelengths l in the range of ~400nm to ~750nm are called visible light.  We see light because it stimulates the cells in our eyes.  Because our eyes are able to distinguish between different wavelength of light we perceive color.  If the light reaching our eyes contains a broad mixture of wavelength, we interpret it as white light.  Because light is an EM wave, it exhibits several behaviors characteristic of waves such as reflection, refraction and diffraction.

In a homogeneous, isotropic medium light travels in a straight line.  When we visually perceive the world around us, we implicitly assume that light follows a straight-line path.  But when light encounters a boundary between two media with different indices of refraction, or when it travels through a non-homogeneous or non-isotropic medium, its path may not be a straight line.  If we neglect diffraction, then we can analyze the propagation of light through different media by analyzing the path of a light ray.  This is called the ray approximation of geometrical optics.

Reflection

Reflection is the abrupt change in the direction of propagation of a wave that strikes the boundary between two different media.  At least some part of the incoming wave remains in the same medium.  Assume the incoming light ray makes an angle q_i with the normal of a plane tangent to the boundary.  Then the reflected ray makes an angle q_r with this normal and lies in the same plane as the incident ray and the normal.

Law of reflection: q_r = q_i
The angle of reflection equals the angle of incidence.

Specular reflection occurs at smooth, plane boundaries.  Then the plane tangent to the boundary is the boundary itself.  Reflection at rough, irregular boundaries is diffuse reflection.  The smooth surface of a mirror reflects light specularly, while the rough surface of a wall reflects light diffusely.

The reflectivity of a surface material is the fraction of energy of the oncoming wave that is reflected by it.  The reflectivity of a mirror is close to 1.

Problem:

Refraction

Refraction is the change in direction of propagation of a wave when the wave passes from one medium into another, and changes its speed.  Light waves are refracted when crossing the boundary from one transparent medium into another because the speed of light is different in different media.  Assume that light waves encounter the plane surface of a piece of glass after traveling initially through air as shown in the figure below.

What happens to the waves as they pass into the glass and continue to travel through the glass?  The speed of light in glass or water is less than the speed of light in a vacuum or air.  The speed of light in a given substance is v = c/n, where n is the index of refraction of the substance.  The index of refraction is a material property and in general depends on the wavelength of the light wave.  Typical values for the index of refraction of glass are between 1.5 and 1.6, so the speed of light in glass is approximately two-thirds the speed of light in air.  The distance between wavefronts will therefore be shorter in the glass than in air, since the waves travel a smaller distance per cycle.

If f is the frequency of the wave and t = 1/f is the period, i.e. the time interval between successive crests passing a fixed point in space, then l₁ = v₁t = ct/n₁ and l₂ = v₂t = ct/n₂, or

l₁/l₂ = n₂/n₁.

Now consider wavefronts and their corresponding light rays approaching the surface at an angle.

We can see that the rays will bend as the waves pass from air to glass.  The bending occurs because the wavefronts do not travel as far in one cycle in the glass as they do in air.  As the diagram shows, the wavefront halfway into the glass travels a smaller distance in glass than it does in air, causing it to bend in the middle.  Thus, the ray, which is perpendicular to the wavefront, also bends.  The situation is like a marching band marching onto a muddy field at an angle to the edge of the field.  The rows bend as the speed of the marchers is reduced by the mud.  The amount of bending depends on the angle of incidence and on the indices of refraction of glass and air, which determine the change in speed.  From the figure we can see that

l₁/l₂ = sinq₁/sinq₂.

But l₁/l₂ = n₂/n₁.  Therefore

n₂/n₁ = sinq₁/sinq₂, or n₁sinq₁ = n₂sinq₂.

This is Snell's law, or the law of refraction.

n_isinq_i = n_tsinq_t.

When light passes from one transparent medium to another, the rays are bent toward the surface normal if the speed of light is smaller in the second medium than in the first.  The rays are bent away from this normal if the speed of light in the second medium is greater than in the first.  The picture below shows a light wave incident on a slab of glass.

One part of the wave is reflected, and another part is refracted as it passes into the glass.  The rays are bent towards the normal.  At the second interface from glass into air the light passing into the air is refracted again.  The rays are now bent away from the normal.

Problem:

Total internal reflection

When light propagates from air into glass or from glass in to air it may change its direction of travel. Snell's law reveals the relationship between the directions of travel in the two media.

n₁sinq₁ = n₂sinq₂

Consider light propagating in glass with index of refraction n₁ = 1.5 towards a glass-air boundary.  If the angle the light makes with the normal to the boundary in the glass is q₁, then the angle it makes in the air is given by

sinq₂ = (n₁/n₂)sinq₁ = 1.5 sinq₁.

If sinq₁ > (1/1.5) = 2/3, or q₁ > 41.8^o, then sinq₂ is greater than 1 and there is no solution for q₂.  The angle q_c for which sinq_c = n₂/n₁ is called the critical angle.  For angles greater than the critical angle there exists no solution for q₂, and there is no refracted ray.  The incident light is totally reflected, obeying the law of reflection.  If n₂ = 1.5 and n₁ = 1 then the critical angle is q_c = 41.8^o.

Total internal reflection occurs only if light travels from a medium of high index of refraction to a medium of low index of refraction.  Let light travel from medium 1 into medium 2 and let n₁ > n₂.  Then the critical  angle q_c is given by

sinq_c = n₂/n₁

For angles greater than the critical angle the incident light is totally reflected, obeying the law of reflection.

Dispersion

The velocity of light in a material, and hence the index of refraction of the material, depends on the wavelength of the light.  The index listed in tables is either an average index, or it is the index for one particular wavelength.  Since the refractive index depends on the wavelength of the light, light waves with different wavelengths and therefore different colors are refracted through different angles.  This is called dispersion, because white light is dispersed into its component colors while traveling through the material.  In general, the index of refraction n varies inversely with wavelength.  It is greater for shorter wavelengths.

The table below gives the index of refraction for various wavelengths of light in glass.

Snell's law combined with a wavelength-dependent index of refraction n explains the dispersive properties of a prism.  The sides of a prism are not parallel and light changes direction when it passes through it.

A ~1% variation in the index of refraction over the entire visible range of electromagnetic radiation still results in a significant change in the direction of the emerging red and blue rays.  Since in general the index of refraction is bigger for shorter wavelengths, blue light gets bent more than red light.

Rainbows

A rainbow is produced by dispersion and internal reflection of light in water droplets in the atmosphere.  White light from the sun enters a spherical raindrop.  The different colors are refracted through different angles, reflected off the back of the drop, and then refracted again when they emerge from the drop.  The white light now has been dispersed into its component colors, and the different colors travel in slightly different directions.  You see red light coming from water droplets higher in the sky than violet light.  The other colors are found between these, making a rainbow.  In the figure below, red light arrives at the eye of the observer from the upper drop and violet light from the lower drop. Other raindrops yield the other colors.

Rainbows are usually seen as half circles.  From a plane or from a very tall building or mountain, however, one can see a complete circle.  Sometimes one can see a double rainbow.  The second, dimmer, band, which is higher in the sky than the first, comes from light reflected twice inside a raindrop.  This reverses the order of the colors in the second band.

Sunlight

Sunlight originates at the outer surface of the sun, in a region called the photosphere.  This region has a temperature of ~5800^oC. The distribution of wavelength in sunlight is determined by the temperature of the photosphere.  Not all sunlight is visible.  EM waves in the infrared and ultraviolet part of the EM spectrum are also produced in the photosphere.

Sunlight travels from the Sun to the Earth through empty space with the speed of light c.  When it enters the earth atmosphere it is refracted.  The index of refraction of air near sea level is only 1.0003, so the refraction is barely noticeable.  Air molecules, water molecules and dust also scatter some of the light (Rayleigh scattering).  These particles scatter shorter wavelength light more efficiently that longer wavelength light.  Scattering by tiny particles is always most efficient when the wavelength of the EM wave approximately matches the size of the tiny particle.  The dimensions of the molecules and dust particles are much smaller than the wavelengths of visible light, so blue light with the shortest wavelength provides the best match.  Blue light is scattered more than red light.  Most sunlight travels directly to our eyes, but the scattered light reaches us by a more complicated path from different directions.  We therefore see the brilliant yellow disk of the sun (direct light) and the fairly uniformly blue sky (scattered light).

As the sun rises or sets, light must travel a long distance through the Earth's atmosphere to reach our eyes.  Most of the blue light is scattered away during this passage through the atmosphere and the direct light from the sun appears red.  Extra dust and ash particles from pollution, forest fires, or volcanic eruptions enhance the Rayleigh scattering and are responsible for unusually red sunrises and sunsets.

Clouds and fog are composed of relatively large water droplets, larger than the wavelengths of visible light.  All wavelengths in the visible spectrum are scattered very efficiently by these large particles, so that very little direct sunlight reaches our eyes.  The scattered light, however, does not have any particular color, and clouds and fog appear white.

Fermat's principle

The law of reflection and the law of refraction tell us how light waves behave at the boundary between two media with different indices of refraction.  In 1650, Fermat discovered a way to explain reflection and refraction as the consequence of one single principle.  It is called the principle of least time or Fermat's principle.

Assume we want light to get from point A to point B, subject to some boundary condition.  For example, we want the light to bounce off a mirror or to pass through a piece of glass on its way from A to B.  Fermat's principle states that of all the possible paths the light might take, that satisfy those boundary conditions, light takes the path which requires the shortest time.

Consider the diagram above.  We want light to leave point A, bounce off the mirror, and get to point B.  Let the perpendicular distance from the mirror of both A and B be d and the shortest distance between the points be D.  Assume that light takes the path shown.  The length of this path is

L = (x²+d²)^1/2 + ((D-x)²+d²)^1/2.

Since the speed of light is the same everywhere along all possible paths, the shortest path requires the shortest time.  To find the shortest path, we differentiate L with respect to x and set the result equal to zero.  (This yields an extremum in the function L(x).)

.

After canceling equal terms on both sides we are left with

d²D² = 2Dx,   or x = D/2.

The path that takes the shortest time is the one for which x = D/2, or equivalently, the one for which q_i = q_r.  Fermat's principle yields the law of reflection.

Now assume we want light to propagate from point A to point B across the boundary between medium 1 and medium 2.

For the path shown in the figure above the time required is

.

Setting dt/dx = 0 we obtain

,

or

n₁sinq₁ = n₂sinq₂.

Fermat's principle yields Snell's law.

Please complete assignment 20 now.  For the User ID use your registered password.

You can submit the assignment up to three times.  Each time the computer will tell you your score.