Refraction and Reflection

The index of refraction

The triumph of Maxwell's theory is summarized in its correct prediction of the speed of light in vacuum. The wave equation predicts that the velocity of the wave is

The fundamental constants e₀ (the permittivity of vacuum) and m₀ the (permeability of vacuum) were known and so the speed of light was predicted. Experimental measurement of the speed of light gave good agreement with the theory. Suppose, however, that the electromagnetic wave is propagating in a medium other than vacuum. The constants e and m are not equal to the vacuum values. In general the speed of the wave is

The speed v is always less than c. We can define the index of refraction n as the ratio of c/v. Thus, n = 1 for vacuum and is always greater than 1 for any dielectric. The index of refraction is

For non-magnetic materials m » m₀ and so usually we have

where e_r is the relative permittivity. The relative permittivity is the ratio of the permittivity relative to that in vacuum. The relative permittivity is also called the dielectric constant of the material. It is important to realize that the dielectric constant is frequency dependent. The high frequency response is relevant here because we are considering the interaction of electromagnetic radiation in radiofrequency range of higher for the purposes of this discussion. For example, for 500 nm (green) light the frequency can be calculated as follows.

The wavelength l = 500 nm = 5 x 10^-7 m. The speed of light is given by c = ln where n is the frequency. Thus,

n = c/l = 3 x 10⁸ ms^-1/(5 x 10^-7 m) = 6 x 10¹⁴ s^-1

or 600 teraHertz. This is a quite a high frequency.

Since the dielectric constant is a function of frequency, e(n), the index of refraction is as well, n(n).

The index of refraction is complex and thus can be written, n' = n + ik. The real and imaginary parts correspond to in-phase and out-of-phase components of the frequency response. The in-phase part is responsible for dispersion of light as it enters a medium. This dispersion leads to refraction as we shall see below. The out-of-phase component is responsible for absorption of radiation. Since the absorption of radiation is a central topic in this course this connection will be discussed in detail later. At this stage we wish to make the point that there is a relationship between dispersion (real part of index of refraction) and absorption (imaginary term). Typical absorption and associated dispersion curves are shown in the Figure below.

There are several interesting properties of dielectric materials. First, the propagation speed of light is altered and therefore its path through a dielectric must be altered due to the dielectric constant. Second, at a boundary or interface between two dielectric materials electromagnetic radiation can either be transmitted or reflected.

Normally, we think of blue light being more refracted than red. This means that for many materials the real part of the index of refraction is larger for shorter wavelength (higher frequency) electromagnetic waves. We refer to this as the dispersion (i.e. wavelength dependence) of the index of refraction.

Birefringent materials have two indices of refraction along different crystal axes. These are useful materials because the dispersion changes as a function of the angle of the crystal axes with respect to the polarization of electromagnetic radiation.

Reflection

In this section we show that the angle of reflection q_r is equal to the angle of incidence q_i. We begin with a coordinate system shown in the Figure below. Three waves are considered. We wish to find a relationship between these waves. The conditions are:

All three vectors E_i, E_r, and E_t are identical functions of time.
All three vectors are identical functions of position r_I on the interface.

From condition 1 we know that all three waves must have the same frequency.

w_i = w_r = w_t

To define the waves a function of position we consider the definition of the wave number k = n/l, where n is the index of refraction and l is the wavelength divided by 2p. In medium 1 the waves will propagate with k₁ and in medium 2 with k₂. Thus, from condition 2 we must have, at any point on the interface,

Then

Since the vector r_I lies in the interface, the vector n_i - n_r must be normal to the interface. The tangential components of these two vectors must be equal and their angles with respect to the interface must be equal. Therefore, q_i = q_r.

The law of reflection.

The angle of reflection is equal to the angle of incidence.

Refraction

The three vectors are coplanar and define a plane of incidence shown in the Figure above.

For the transmitted wave we have

The tangential components of k₁n_i and k₂n_t must be equal.

Since k = n/l we have Snell's law.

These laws are general. They apply to any two media. The propagating waves are given by:

Without proof we state Snell’s law which states that n₁sinq₁ = n₂sinq₂ for the angle of light at a dielectric boundary shown below.

The combined laws of reflection and refraction are illustrated in the figure below. Note that all vectors shown are in the plane of incidence.

Polarization

We refer to the polarization at the boundary as s or p. s polarized light has an electric vector parallel to the surface while p polarized light has components parallel and perpendicular to the surface. This is illustrated below. The electric vector of s polarized light is perpendicular to monitor and this is represented by concentric circles.