Electromagnetic waves with wavelengths l in the range of ~400 nm to ~750 nm 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 and interference.

A scheme for thinking about the nature of wave propagation is called Huygen's principle.  Huygen's principle is a geometrical construction that tells us how wavefronts will move, but not why they will move that way.  Each point on a wave front is considered to be a point source for the production of new waves.  In three dimensions, these new waves are spherical waves called wavelets, that propagate outward with the characteristic speed of the wave.  The wavelets emitted by all points on the wave front interfere with each other to produce the traveling wave.  When studying the propagation of light, we can replace any wave front by a collection of sources distributed uniformly over the wave front, radiating in phase.

When light passes through a small opening, comparable in size to the wavelength l of the light, in an otherwise opaque obstacle, the wave front on the other side of the opening resembles the wave front shown below.

The light spreads around the edges of the obstacle.  This is the phenomenon of diffraction.

Light rays are orthogonal trajectories to the wave front of an EM wave with frequency in the visible region.  They are lines normal to the wave front at every point of intersection.  In an isotropic medium, light rays are parallel to the wave vector k.

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

Refraction is the change in direction of propagation of a wave when the wave passes from one medium into another and changes its speed.  The speed of light in a given substance is v = c/n, where n is the index of refraction of the substance.  Light waves are refracted when crossing the boundary from one transparent medium into another because the speed of light is different in different media. 

Snell's law, or the law of refraction: n₁sinq₁ = n₂sinq₂.

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.

At a boundary between two transparent media, light is partially reflected and partially refracted.  The ratio of the reflected irradiance to the incident irradiance is called the reflectance R and the ratio of the transmitted irradiance to the incident irradiance is called the transmittance T.  Energy conservation requires that R + T = 1 (if there is no absorption).

R and T depend on the indices of refraction of the two media n₁ and n₂, the angle of incidence q₁, and the polarization of the incident light.  We distinguish between p-polarization and s-polarization.  Let the plane of incidence contain the normal to the boundary and the incident wave vector k₁.  The electric field vector E₁ is perpendicular to k₁.  If we choose our coordinate system as shown below, then plane of incidence is the xz-plane and E₁ may be written as E₁= E_p+ E_s.  E_p lies in the xz-plane and E_s is perpendicular to the xz-plane, i.e. it points in the ±y-direction.  The electric field of the incident light is a linear superposition of p- and s-polarized fields.

For p-polarized light we have R = |r_12p|², where r_12p is the Fresnel reflection coefficient for p-polarization.  We have  

r_12p = tan(q₁ - q₂)/tan(q₁ + q₂).

For s-polarized light we have R = |r_12s|², where r_12s is the Fresnel reflection coefficient for s-polarization.  We have  

r_12s = sin(q₁ - q₂)/sin(q₁ + q₂).

For a graph of the reflectance R for s- and p-polarized light as a function of n₁, n₂, and q₁, download this Excel spreadsheet.

If q₁ + q₂ = p/2,then tan(q₁ + q₂) = ¥ and r_12p = 0.  If light is reflected, it will have s-polarization.  The incident angle at which this happens is called the Brewster angle q_B.   We then have

n₁sinq_B = n₂sin((p/2) - q_B) = n₂cosq_B.

tanq_B = n₂/n₁.

Polarized light can thus be obtained via reflection.

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.