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This subchapter can easily be turned into a whole lecture course, so it is impossible
to derive all the interesting relations and to cover anything in depth. This subchapter
therefore just tries to give a strong flavor of the topic. |
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We know, of course,
that the index of refraction
n
of a non-magnetic material is linked to the dielectric constant er
via a simple relation, which is a rather direct result of the Maxwell equations. |
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But in learning about the origin of the dielectric constant, we have progressed from a simple
constant er to a complex dielectric function
with frequency dependent real and imaginary parts. |
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What happens to n then? How do we transfer the wealth of additional information contained
in the dielectric function to optical properties, which are to a large degreee encoded
in the index of refraction? |
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Well, you probably guessed it: We switch to a complex
index of refraction! |
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But before we do this, let's ask ourselves what we actually want to find out. What are the optical properties
that we like to know and that are not contained in a simple index of refraction? |
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Lets look at the paradigmatic experiment in optics and see what we
should know, what we already know, and what we do not yet know. |
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What we have is an electromagnetic wave, an incident
beam (traveling in vacuum to keep things easy), which impinges on our dielectric material. As a result we obtain a reflected beam traveling in vacuum and a refracted beam which
travels through the material. What do we know about the three beams? |
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The incident beam is characterized by its wavelength li,
its frequency ni and its velocity c0, the direction of its
polarization in some coordinate system of our choice, and the arbitrary angle of incidence
a. We know, it is hoped, the simple dispersion
relation for vacuum. |
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c0 is, of course, the velocity of light in vacuum, an absoute constant of nature. |
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The
incident beam also has a certain amplitude of the
electric field (and of the magnetic field, of course) which we call E0. The intensity
Ii of the light that the incident beams embodies, i.e. the energy flow, is proportional to E02
- never mix up the two! |
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The reflected beam follows one of the basic laws of optics,
i.e. angle of incidence = angle of emergence, and its wavelength, frequency and magnitude
of velocity are identical to that of the incident beam. |
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What we do not know is its amplitude
and its polarization, and these two quantities must somehow depend on the properties
of the incident beam and the properties of the dielectric. |
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If we now consider the refracted beam, we know that it
travels under an angle b, has the same frequency as the incident beam, but a wavelength
ld
and a velocity c that is different from li
and c0. |
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Moreover, we must expect that it is damped or attenuated
, i.e. that its amplitude decreases as a function of penetration depth (this is indicated by decreasing thickness of
the arrow above). All parameters of the refracted beam may depend on the polarization of the incident beam. |
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Again, basic optics teaches that there are some simple
relations. We have
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| | Snellius
law | | |
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| | From Maxwell equations |
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| c = ni · ld
| | Always valid | | |
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| | From the equations above |
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A bit more involved is another basic relations coming from the Maxwell equations. It is the equation linking
c, the speed of light in a material to the material "constants" er
and the corresponding magnetic permeability m0 of vacuum and mr
of the material via |
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| c = |
1
(m
0 · mr · e 0 · er)1/2 |
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Since most optical materials are not magnetic, i.e. mr
= 1, we obtain for the index of refraction of a dielectric material our relation from above. |
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| n | = |
c 0
c | =
|
(m0 · m r · e0 · er )1/2
(m0 · e0)1/2 |
= |
er1/2 |
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Consult the basic optics module if you have problems
so far. |
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If we now look at not-so-basic optics, we encounter the Fresnel laws of diffraction. |
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Essentially, the Fresnel laws give the
intensity of the reflected beam as a function of the angle of incidence
, the polarization of the incident beam, and the index
of refraction of the material. |
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The Fresnel laws are not particularly easy to obtain (consult the basic module Fresnel
laws), but the results are easy. |
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First, we must distinguish between the two basic polarization cases possible: |
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The incident light might be polarized in such a way that the vector of the electrical field E
lies either in the plane of the material, or perpendicular
to it, as shown below. Anything in between than can be decomposed into the two basic cases. |
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Lets call the amplitudes of the reflected beam Apara for the case of the polarization
being parallel to the plane (= surface of the dielectric), and Aperp
for the case of the polarization being perpendicular
to the plane (blue case) as shown above. For a unit amplitude of the incident beam, the Fresnel laws then state |
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| Apara = – |
sin(a – b)
sin( a + b) |
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Aperp = – |
tan(a – b)
tan( a + b) |
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We can substitute the angle b
by using the relation from above and the resulting equations then give the intensity
of the reflected light as a function of the material parameter n . Possible, but the resulting equations are
no longer simple. |
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In order to stay simple and focus on the essentials, we will now consider only cases with
about perpendicular incidence, i.e. a
» 0o. This makes everything much easier. |
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At small angles we may substitute the argument of the
sin or tan for the full function, and obtain for both polarizations |
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Using the expression for n from above for small angles too, we obtain |
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Now we keep in mind that we are usually interested in intensities
, and not in amplitudes. Putting everything together, we obtain for the reflectivity R , defined as the ratio
of the intensity Ir of the reflected beam to the intensity Ii of the incident
beam for almost perpendicular incidence |
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| R = |
Ir
I i | = |
(n – 1)2
(n + 1)2 |
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The grand total of all of this is that if we know n and some basics
about optics, we can answer most, but not all of the questions from
above. But so far we also did not use a complex index of refraction either. |
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In essence, what is missing is any statement about the attenuation
of the refracted beam, the damping of the light inside the dielectric - it is simply
not contained in the equations presented so far. |
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This cannot be right. Electromagnetic radiation does not penetrate arbitrarily thick (and
still perfect) dielectrics - it gets pretty dark, for example, in deep water even if it is perfectly clear. |
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In not answering the "damping" question, we even raise a new question: If we include
damping in the consideration of wave propagation inside a dielectric, does it change the simple equations given above? |
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The bad news is: It does! But relax: The good news is: |
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All we have to do is to exchange the "simple" refractive index n by
a complex refractive index
n* that is directly tied to the complex dielectric function, and everything is taken care of.
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We will see how this works in the next paragraph. |
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© H. Föll (Electronic Materials - Script)
