2.4: Classical Light-Waves
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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:
Ey(x,t)=Re[ψ(x,t)]=|ˉψ|cos(kx−ωt+φ),
where |ˉψ| is the amplitude of the electric oscillation, k the wavenumber, ω the angular frequency, and φ the phase angle. In addition, λ=2π/k is the wavelength, and ν=ω/2π 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=νλ,
Ey(x,t)=|ˉψ|cos(k[x−(ω/k)t]+φ)=|ˉψ|cos(k[x−ct]+φ).
Note that Ey depends on x and t only via the combination x−ct. It follows that the wave maxima and minima satisfy x−ct=constant.
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
vp=ωk,
From standard electromagnetic theory , the energy density (i.e., the energy per unit volume) of a plane light-wave is
U=E2yϵ0,
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)