# Refractive Index and Absorption

• Heinrich Hora

## Abstract

As derived from the macroscopic theory of a plasma, the complex optical refractive index ñ is given by the dispersion relation of electromagnetic waves in a plasma, (Eq.:(5.12)), where the real part n, and the imaginary part κ, are evaluated algebraically
$$\tilde{n}=n+i\kappa ={{(1-\frac{{{\omega }_{p}}^{2}}{\omega (1+i\nu /\omega )})}^{1/2}}$$
(6.1)
$$n={{\left[ \frac{1}{2}\left\{ \left. {{\left[ {{\left( 1-\frac{{{\omega }_{p}}^{2}}{{{\omega }^{2}}+{{\nu }^{2}}} \right)}^{2}}+{{\left( \frac{\nu }{\omega }-\frac{{{\omega }_{p}}^{2}}{{{\omega }^{2}}+{{\nu }^{2}}} \right)}^{2}} \right]}^{1/2}}+\left( 1-\frac{{{\omega }_{p}}^{2}}{{{\omega }^{2}}+{{\nu }^{2}}} \right) \right\} \right. \right]}^{1/2}}$$
(6.2)
$$\kappa ={{\left[ \frac{1}{2}\left\{ \left. {{\left[ {{\left( 1-\frac{{{\omega }_{p}}^{2}}{{{\omega }^{2}}+{{\nu }^{2}}} \right)}^{2}}+{{\left( \frac{\nu }{\omega }-\frac{{{\omega }_{p}}^{2}}{{{\omega }^{2}}+{{\nu }^{2}}} \right)}^{2}} \right]}^{1/2}}-\left( 1-\frac{{{\omega }_{p}}^{2}}{{{\omega }^{2}}+{{\nu }^{2}}} \right) \right\} \right. \right]}^{1/2}}$$
(6.3)
The real part n is sometimes called the refractive index. Here the sum with the complex refractive index is denoted by the circumflex ñ. For a collisionless plasma (υ = 0), both coefficients are clearly equivalent
$$n=\tilde{n}={{(1-{{\omega }_{p}}^{2}/{{\omega }^{2}})}^{1/2}}\ (if\nu =0)$$
(6.4)
The imaginary part of ñ, κ, is called the absorption coefficient. Its meaning is seen immediately from its relation to the absorption constant K, which determines the attenuation of a laser intensity I at some depth x; if I0 is the intensity at x = 0
$$\operatorname{I} = {I_0}{\text{exp}}( - \operatorname{Kx}$$
(6.5)
The absorption constant is then
$${\rm K} = \frac{{2\omega }}{c}\kappa$$
(6.6)

## Keywords

Refractive Index Laser Intensity Collision Frequency Oscillation Energy Coulomb Collision