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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. |
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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 |
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| Ex |
= E0, x · exp i ·
(kx · x – w · t) |
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With kx = component of
the wave vector in x-direction = k = 2p/l, w = circular frequency = 2pn. |
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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. |
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In other words, we use |
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Of course, c is the speed of light in
vacuum. Insertion yields |
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| Ex |
= |
E0, x · exp i · |
æ
ç
è |
w · n*
c |
· x – w
· t |
ö
÷
ø |
= |
E0, x · exp i · |
æ
ç
è |
w · (n + i ·
k)
c |
· x – w ·
t |
ö
÷
ø |
| Ex |
= |
E0, x · exp · |
æ
ç
è |
i · w · n
· x
c |
– |
w · k ·
x
c |
– i · w ·
t |
ö
÷
ø |
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The red expression is nothing but the wavevector,
so we get a rather simple result: |
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| Ex |
= exp – |
w · k ·
x
c |
· exp[ i · (kx · x
– w · t)] |
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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. |
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Obviously, at a depth often called absorption
length or penetration depth W = c/w ·
k, the intensity decreased by a factor
1/e. |
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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 . |
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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. |
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(to be
continued) |
© H. Föll
