Light propagation in absorbing media

Light propagation in absorbing media

An absorbing medium is characterised by the fact that the energy of the EM radiation
is dissipated in it. This would imply that the amplitude of a plane EM wave decreases
exponentially as the wave propagates along the absorbing medium. The mathematical
description of light propagation in absorbing media can be treated by considering that
the dielectric permittivity is no longer a real number, but a complex quantity εc. In
terms of field description, this implies that the electric displacement, now related by
the electric field by D = εcE, will not be in phase with the electric field in general.
As the refractive index was defined as a function of the dielectric permittivity, it will
be in general a complex number itself, now defined by:


where nc is called the complex refractive index. It is useful to work with the real and
imaginary part separately, and in this way we define:
nc = n − iκ (2.71)
where now n is the real refractive index, and κ is called the absorption index.


In addition, from the Helmholtz equation (2.44) the relation between the complex
wavevector kc (now complex) and the complex refractive index nc is
k2c
≡ ω2εcμ = n2c
k0 (2.72)
Because the wavevector is now a complex vector, we can separate its real and
imaginary parts in the following way:
kc ≡ k − ia (2.73)
where k represents the real wavevector, and a is called the attenuation vector. The
relation between the vectors k and a and the optical constant of the material medium
n and κ are deduced from equation (2.72), resulting in
k2 − a2 = k2
0(n2 − κ2) (2.74)
ka = k20nκ


The planes of constant amplitude will be determined by the condition ar = constant,
and therefore they will be planes perpendicular to the attenuation vector a. On the other
hand, the planes of equal phase will be defined by the condition of kr = constant, and
thus the phase front will be planes perpendicular to the real wavevector k. In general,
these two planes will not be coincident, and in this case the EM wave is said to be an
inhomogeneous wave.
Nevertheless, the most common situation faced in light propagation in absorbing
media is the case where the vectors k and a are parallel, and such a wave is called a
homogeneous wave. In this particular case, the vectors kc, k and a are related to the
optical constant of the medium through the following simple relations:
k = nk0 (2.77)
a = κk0 (2.78)
kc ≡ (n − iκ)k0