3.7.2 Using the Complex Index of Refraction

3.7.3 Using the Complex Index of Refraction

Lets look at the physical meaning of n and k, i.e. the real and complex part of the complex index of refraction, by looking at an electromagnetic wave traveling through a medium with such an index.

For that we simply use the general formula for the electrical field strength E of an electromagnetic wave traveling in a medium with refractive index n*. For simplicities sake, we do it one-dimensional in the x-direction (and use the index "x" only in the first equation). In the most general terms we have

E_x	= E_{0, x} · exp i · (k_x · x – w · t)

With k_x = component of the wave vector in x-direction = k = 2p/l, w = circular frequency = 2pn.

No index of refraction in the formulas; but we know (it is hoped), what to do. We must introduce the velocity v of the elecromagnetic wave in the material and use the relation between frequency, wavelength, and velocity to get rid of k or l, respectively.

In other words, we use

v	=	c n*			v	=	n · l

k	=	2p l	=	w · n* c

Of course, c is the speed of light in vacuum. Insertion yields

E_x	=	E_{0, x} · exp i ·	æ ç è	w · n* c	· x – w · t	ö ÷ ø	=	E_{0, x} · exp i ·	æ ç è	w · (n + i · k) c	· x – w · t	ö ÷ ø

E_x	=	E_{0, x} · exp ·	æ ç è	i · w · n · x c	–	w · k · x c	– i · w · t	ö ÷ ø

The red expression is nothing but the wavevector, so we get a rather simple result:

E_x	= exp –	w · k · x c	· exp[ i · (k_x · x – w · t)]

In words that means: if we use a complex index of refraction, the propagation of electromagnetic waves in a material is whatever it would be for a simple real index of refractions times a damping factor that decreases the amplitude exponentially as a function of x.

Obviously, at a depth often called absorption length or penetration depth W = c/w · k, the intensity decreased by a factor 1/e.

The imaginary part k of the complex index of refraction thus describes rather directly the attenuation of electromagnetic waves in the material considered. It is known as damping constant, attenuation index, extinction coefficient, or (rather misleading) absorption constant. Misleading, because an absorption constant is usually the a in some exponential decay law of the form I = I₀ · exp – a · x .

Note: Words like "constant", "index", or "coefficient" are also misleading - because k is not constant, but depends on the frequency just as much as the real and imaginary part of the dielectric function.

(Should be continued but won't)