Monochromatic waves

Monochromatic waves

The time dependence of the electric and magnetic fields within the wave equations

admits solutions of the form of harmonic functions. Electromagnetic waves with such

sinusoidal dependence on the time variable are called monochromatic waves, and are

characterised by their angular frequency ω. In a general form, the electric and magnetic

fields associated with a monochromatic wave can be expressed as:

E(r, t) = E0(r) cos[ωt + ϕ(r)] 

H(r, t) = H0(r) cos[ωt + ϕ(r)]

where the fields amplitudes E0(r) and H0(r) and the initial phase ϕ(r) depend on

the position r, but the time dependence is carried out only in the cosine argument

through ωt.

When dealing with monochromatic waves, in general it is easier to write down the

monochromatic fields using complex notation. Using this notation, the electric and

magnetic fields are expressed as:

E(r, t) = Re[E(r)e+iωt ]

H(r, t) = Re[H(r)e+iωt ]

where E(r) and H(r) denote the complex amplitudes of the electric and magnetic

fields, respectively (see Appendix 1). The angular frequency ω that characterises the

monochromatic wave is related to the frequency ν and the period T by:

ω = 2πν = 2π/T 

The electromagnetic spectrum covered by light (optical spectrum) ranges from frequencies

of 3 × 105 Hz corresponding to the far IR, to 6 × 1015 Hz corresponding to

vacuum UV, being the frequency of visible light around 5 × 1014 Hz.

The average of the Poynting vector as a function of the complex fields amplitudes

for monochromatic waves takes the form:

S = Re{Ee+iωt} × Re{He+iωt } = Re{S} (2.37)

where S has been defined as:

S = 1/2 E × H∗

(2.38)

and is called the complex Poynting vector. In this way, the intensity carried by a

monochromatic EM wave should be expressed as:

I = |Re{S}| (2.39)

In the case of monochromatic waves, Maxwell’s equations using the complex fields

amplitudes E and H are simplified notably, because the partial derivatives in respect

of the time are directly obtained by multiplying by the factor iω:

∇E = 0 (2.40)

∇H = 0 (2.41)

∇ ×E = −iμ0ωH 

∇ ×H = iεωE

where we have assumed a dielectric and non-magnetic medium in which σ = 0

and μ = μ0.

Now, if we substitute the solutions on the form of monochromatic waves (2.34) and

(2.35) in the wave equation (2.24), we obtain a new wave equation, valid only for

monochromatic waves, known as the Helmholtz equation:

∇2U(r) + k2U(r) = 0 (2.44)

where now U(r) represents each of the six Cartesian components of the E(r) and H(r)

vectors defined in (2.34) and (2.35), and where we have defined k as:

k ≡ ω(εμ0)1/2 = nk0 (2.45)

k0 ≡ ω/c (2.46)

If the material medium is inhomogeneous the dielectric permittivity is no longer

constant, but position dependent ε = ε(r). In this case, although Maxwell’s equations

remain valid, the wave equation (2.24) or the Helmholtz equation (2.44) are not longer

valid. Nevertheless, for a locally homogeneous medium, in which ε(r) varies slowly

for distances of ∼1/k, those wave equations are approximately valid by now defining

k = n(r)k0, and n(r) = [ε(r)/ε0)]1/2.