The absorption coefficient is therefore described as the reciprocal of the depth of penetration of radiation into a bulk solid, i.e., it is equal to the depth at which the energy of the radiation has decreased by the factor of e -αx, or alternatively, the intensity of the incident radiation is attenuated by the solid to 1/e of its initial value at a distance from the surface boundary defined by λ /4πk. This equation is known as Bouguer's law or Lambert's law of absorption, by which radiation is absorbed to an extent that depends on the wavelength of the radiation and the thickness and nature of the medium. Is a measure of the damping factor, or extinction coefficient ( k).Īs the power ( P) or intensity of an incident wave through a solid is the conductivity (σ) of the solid multiplied by the square of the electric field vector ( P=σE 2), then using the damping factor term, the fraction of the incident power that has propagated from position ( o) to a distance ( x) through the material with conductivity (σ) is given by:įrom which the absorption coefficient (α) can be expressed in terms of the extinction coefficient ( k) as:Īs the velocity of light in a vacuum, c = fλ, then α = 4πk/λ, and the power or intensity is P = P oexp -αx. Therefore, substituting 1/ V into equation above produces: The velocity of propagation through the solid of complex refractive index N = n - ik is related to the speed of light in a vacuum, c, by V = C / N, then:
Is the displacement at time t after a disturbance, created by the electric field at a point situated at x along the line of propagation.įrom Maxwell's equations on electromagnetic theory, the speed of light in a vacuum c is related to the permittivity of free space ε 0, (the degree to which a medium can resist the flow of charge, defined by the ratio of the electric displacement to the intensity of the electric field that produces it), and the permeability of free space µ 0 (the ratio of the magnetic flux density in a solid to the external magnetic field strength inducing it, µ = B/H.) by the equation c=1/(µ 0ε0)½. Where, ( E 0)is the incident electric field vector and
If a plane wave of frequency ( f) propagates through a solid with velocity ( v) in a direction defined by ( x), the electric field ( E) is described by the following progressive wave equation: The optical properties of the solid are therefore governed by the interaction between the solid and the electric field of the electromagnetic wave. The velocity of propagation of a electromagnetic wave through a solid is given by the frequency-dependent complex refractive index N = n - ik where the real part, n is related to the velocity, and k, the extinction coefficient is related to the decay, or damping of the oscillation amplitude of the incident electric field. Absorption and extinction coefficient theory