Refractive Index and Absorption

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)