Classical Description of Electromagnetic Radiation

The wave picture of electromagnetic radiation results from Maxwell's equations. These equations are fundamental postulates that are valid in both classical and quantum physics. In free space, they are:

These equations apply in vacuum. The expression shown are the divergence in the first two equations and the curl in the last two equations. The divergence is

The curl is

where i, j, and k are unit vectors in the x, y, and z directions, respectively. Thus, the divergence is a scalar, but the curl is a vector. Note that the direction of the curl follows the right hand rule. For example, for current flow down a wire, the magnetic field is perpendicular to the direction of current flow and wraps around the direction as your fingers wrap around an extended hitchhiker’s thumb.

In the presence of a charge density r we have that

The divergence of the magnetic induction B is always zero since there is no magnetic monopole. A charge is considered to be an electric monopole and r is the charge density.

Maxwell's equations can be recast as wave equations. They have the following form.

These are classical wave equations where the velocity of electromagnetic radiation is c = (e₀m₀)^-1/2. The Maxwell equations show that there is symmetrical relationship between electric and magnetic fields. It is often convenient to express these equations in terms of the scalar potential f and the vector potential A. Both E and B can be derived from these. The expressions are:

We can add the gradient of any function of r to A without changing E or B. In the Coulomb gauge

The vector potential also obeys the classical wave equation

A solution to the wave equation is

where e (carnet) is a unit vector in a particular direction of space. In free space

The electric field is thus

Since

the equation

requires that the propagation vector be perpendicular to the scalar potential and thus to E. This is a statement that the Poynting vector (or propagation vector) of the light is perpendicular to the electric field. Furthermore,

The electric and magnetic fields are in phase and have the same frequency. The magnetic vector is perpendicular to both the electric vector and the propagation direction. From these considerations we can construct the following picture of an electromagnetic wave:

Where E give the direction of the electric vector, B the direction of the magnetic vector and S the propagation vector.

The relative amplitudes of the electric and magnetic vectors can also be determined to be

It is generally true that E₀ = cB₀ for all electromagnetic waves. The small magnitude of B₀ relative to E₀ will turn out to be important. For example, it explains why magnetic dipole transitions are orders of magnitude weaker than electric dipole transitions.

The energy density is

The intensity or irradiance is I = uc. It is the energy per unit time per unit area. The Poynting vector S is defined to point in the direction of propagation and has a magnitude equal to I.

Polarization properties of light

Linearly polarized light such as that shown in the Figure below can be represented by

The coordinate system is such that z is the vertical axis, y is in the plane of the monitor and x is pointed towards the viewer.

The above electromagnetic wave is z polarized. Note that our convention is to refer to the polarization of the electric field vector of the light. Linearly polarized light is often produced by a laser. The reason for this is frequently the Brewster's angle cut (see refraction below) of the windows in the tube or crystal if the laser is a solid state laser. If both z and x polarizations are present with equal amplitude then the light is unpolarized. Note that this is an example of superposition. The electric vectors in the x and z directions can be rotated by an arbitrary angle about the y axis to give the same polarization.

The purple, blue or red combination gives the same result as the original black. Another way to say this is that one can always project an arbitrary electric vector onto the x and z axes so that the net polarization depends on the relative magnitude of the electric vector and x and z axes. Note that both polarizations are in phase.

The light that emerges from source such as a tungsten-halogen bulb or a xenon arc lamp is frequently unpolarized.

If the x and z electric vectors are not in phase then the light is elliptically polarized.

If the phase angle happens to be 90^o and the amplitudes E_x0 and E_z0 are equal then the light is circularly polarized. For circularly polarized light the electric vector appears to rotate when viewed along the Poynting vector. For clockwise rotation of the electric vector (when viewed towards the source) we speak of right-handed circularly polarized light. For counter-clockwise rotation we speak of left-handed circularly polarized light.