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# 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:

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

$\label{e2.2} \bar{\psi} = |\bar{\psi}|\,{\rm e}^{\,{\rm i}\,\varphi},$ where $$\varphi$$ is real. It follows that the physical electric field takes the form

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

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

$c = \nu\,\lambda,$ or, equivalently,

$\label{e2.7} \omega = k\,c,$ where $$c=3\times 10^8\,{\rm m/s}$$ is the velocity of light in vacuum. Equations (2.4.3) and (2.4.5) yield

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

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 $x - c\, t = {\rm constant}.$ Thus, the wave maxima and minima propagate in the $$x$$-direction at the fixed velocity

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

An expression, such as Equation (2.4.5), 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 (2.4.6), the maxima and minima of a plane-wave propagate at the characteristic velocity

$v_p = \frac{\omega}{k},$ which is known as the phase-velocity. Hence, the dispersion relation (2.4.5) 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

$U = \frac{E_y^{\,2}}{\epsilon_0},$ where $$\epsilon_0= 8.85\times 10^{-12}\,{\rm F/m}$$ is the electrical permittivity of free space. Hence, it follows from Equations (2.4.1) and (2.4.3) that

$\label{e2.10} U \propto |\psi|^{\,2}.$ 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

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