2.4: Classical Light-Waves

Consider a classical, monochromatic, linearly-polarized, plane light-wave, propagating through a vacuum in the \(x\)-direction. It is convenient to characterize a light-wave (which is, of course, a type of electromagnetic wave) by specifying its associated electric field. Suppose that the wave is polarized such that this electric field oscillates in the \(y\)-direction. (According to standard electromagnetic theory, the magnetic field oscillates in the \(z\)-direction, in phase with the electric field, with an amplitude which is that of the electric field divided by the velocity of light in vacuum. ) Now, the electric field can be conveniently represented in terms of a complex wavefunction:

\[ \psi (x,t) = \bar{\psi}\,\rm e^{\,{\rm i}\,(k\,x-\omega\,t)}. \label{e2.1} \]

Here, \({\rm i} = \sqrt{-1}\), \(k\) and \(\omega\) are real parameters, and \(\bar{\psi}\) is a complex wave amplitude. By convention, the physical electric field is the real part of the previous expression. Suppose that

\[ \bar{\psi} = |\bar{\psi}|\,\rm e^{\,{\rm i}\,\varphi}, \label{e2.2} \]

where \(\varphi\) is real. It follows that the physical electric field takes the form

\[ E_y(x,t) = {\rm Re}[\psi(x,t)] = |\bar{\psi}|\,\cos(k\,x-\omega\,t +\varphi), \label{e2.3} \]

where \(|\bar{\psi}|\) is the amplitude of the electric oscillation, \(k\) the wavenumber, \(\omega\) the angular frequency, and \(\varphi\) the phase angle. In addition, \(\lambda=2\pi/k\) is the wavelength, and \(\nu=\omega/2\pi\) the frequency (in hertz).

According to standard electromagnetic theory , the frequency and wavelength of light-waves are related according to the well-known expression

\[c = \nu\,\lambda, \nonumber \]

or, equivalently,

\[ \omega = k\,c, \label{e2.7} \]

where \(c=3\times 10^8\,{\rm m/s}\) is the velocity of light in vacuum. Equations \ref{e2.3} and \ref{e2.7} yield

\[ E_y(x,t) =|\bar{\psi}|\,\cos\left(k\,[x-(\omega/k)\,t]+\varphi\right)= |\bar{\psi}|\,\cos\left(k\,[x-c\,t]+\varphi\right). \label{e2.8} \]

Note that \(E_y\) depends on \(x\) and \(t\) only via the combination \(x-c\,t\). It follows that the wave maxima and minima satisfy

\[x - c\, t = {\rm constant}. \nonumber \]

Thus, the wave maxima and minima propagate in the \(x\)-direction at the fixed velocity

\[ \frac{dx}{dt} = c. \label{e2.7a} \]

An expression, such as Equation \ref{e2.7}, that determines the wave angular frequency as a function of the wavenumber, is generally termed a dispersion relation. As we have already seen, and as is apparent from Equation \ref{e2.8}, the maxima and minima of a plane-wave propagate at the characteristic velocity

\[v_p = \frac{\omega}{k}, \nonumber \]

which is known as the phase-velocity. Hence, the dispersion relation \ref{e2.7} is effectively saying that the phase-velocity of a plane light-wave, propagating through a vacuum, always takes the fixed value \(c\), irrespective of its wavelength or frequency.

From standard electromagnetic theory , the energy density (i.e., the energy per unit volume) of a plane light-wave is

\[U = \frac{E_y^2}{\epsilon_0}, \nonumber \]

where \(\epsilon_0= 8.85\times 10^{-12}\,{\rm F/m}\) is the electrical permittivity of free space. Hence, it follows from Equations \ref{e2.1} and \ref{e2.3} that

\[ U \propto |\psi|^2. \label{e2.10} \]

Furthermore, a light-wave possesses linear momentum, as well as energy. This momentum is directed along the wave’s direction of propagation, and is of density

\[ G = \frac{U}{c}. \label{e2.11} \]