Monochromatic plane waves in dielectric media

Monochromatic plane waves in dielectric media

Once the temporal dependence of the electromagnetic fields has been established in
terms of monochromatic waves, let us now consider the spatial dependence of the
fields. For monochromatic waves, the solution for the spatial dependence, carried by


the complex amplitudes E(r) and H(r), can be obtained by solving the Helmholtz
equation (2.44). One of the easiest and most intuitive solutions for this equation,
and also the most frequently used in optics, is the plane wave. The plane wave is
characterised by its wavevector k, and the mathematical expressions for the complex
amplitudes are:


E(r) = E0e−ikr (2.47)
H(r) = H0e−ikr


where the magnitudes E0 and H0 are now constant vectors. Each of the Cartesian components
of the complex amplitudes E(r) and H(r) will satisfy the Helmholtz equation,
providing that the modulus of the wavevector k is:


k = nk0 = (ω/c)n (2.49)
where ω is the angular frequency of the EM plane wave and n is the refractive index
of the medium where the wave propagates.
As the solution given by the electric and magnetic complex amplitudes must satisfy
Maxwell’s equation, by substituting equations (2.47) and (2.48) into (2.42) and (2.43)
the following relations are straightforwardly obtained:
k × H0 = −ωεE0 (2.50)
k × E0 = ωμ0H0 (2.51)
These two formulae, valid only for plane monochromatic waves, establish the relationship
between the electric field E, the magnetic field H and the wavevector k of the
plane wave. From equation (2.50) one obtains that the electric field is perpendicular to
the magnetic field and the wavevector. In the same way, the relation (2.51) establishes
that the magnetic field is perpendicular to E and k. Therefore, one can conclude that
k, E and H are mutually orthogonal, and because E and H lie on a plane normal to the
propagation direction defined by k, such wave in called a transverse EM wave (TEM)
(Figure 2.1).
The fact that these three vectors are perpendicular implies (from equations (2.50)
and (2.51)) that H0 = (ωε/k)E0 and H0 = (k/ωμ0)E0. These two relations can be


simultaneously fulfilled only if the wavevector modulus is k = ω(εμ0)1/2 = ω/v =
nk0. Of course, this is the condition needed for the wave solution described by (2.47)
and (2.48) to fulfil the Helmholtz equation (2.44).
When dealing with a monochromatic plane EM wave it is useful to characterise it by
its radiation wavelength λ, defined as the distance between the two nearest points with
equal phase of vibration, measured along the propagation direction. The wavelength is
therefore expressed by:
λ ≡ vT = v/ν = 2π/k = 2π/nk0 = λ0/n (2.52)
where λ0 represents the wavelength of the EM wave in free space, given by:
λ0 = cT = c/ν = 2π/k0 (2.53)
It is worth remarking that when an EM wave passes from one medium to another
its frequency remains unchanged, but as its phase velocity is modified due to its
dependence on the refractive index, the wavelength associated with the EM wave
should also change. Therefore, when the wavelength of an EM wave is given, it is
usually referred to the wavelength of that radiation propagating through free space.