Prsentation PowerPoint

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Propriétés optiques des solides G. Bastard, G.
Cassabois, R. Ferreira, Ch. Voisin
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Organisation 26/09 GB1 GB2 03/10 GB3
GB4 10/10 TD x 2 17/10 RF1 TD 24/10 RF2
RF3 31/10 TD x 2 07/11 TD x 2 14/11 GC1
GC2 21/11 ChV1 ChV2
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Summary electrodynamics
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In continuous media there exists a very large
number of charges/dipoles or of currents carried
by mobile charges.
As a result, the fields vary strongly over small
distances
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Microscopic Maxwell equations
Too complicated and unmanageable
Clear cut link between charges/currents and fields
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• Impossible to follow all these currents, charges
  etc..
• Besides the wavelength of light l is considerably
  larger than the typical atomic distances
• We shall study the coupling between e- m waves
  and coarsed grained charge and current
  distributions
• SPATIAL AVERAGING

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d
d
Charge and current densities are averaged over
distances d gtgta but d ltlt l
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For example
Normalization lt1gt 1
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Averaged charge density
Conclusion The averaged charge density is
smaller and smoother than the bare one
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The price to pay No longer a clear cut link
between the charge/current densities and the
fields How to write equations of motions for
averaged charge/densities by averaged fields???
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Macroscopic Maxwell equations
Polarization density
Magnetization density

Material  constitutive  relationships between

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For instance, linear relationships at fixed
angular frequency
dielectric susceptibility (in principle c ltlt 1)
Relative dielectric constant (a c - number)
Models are needed to compute c Drude -
Lorentz Improved (quantum) Drude Lorentz
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It is important to be fully aware of the
dynamical coupling between the charges and the
fields. An e-m wave that propagates in a material
medium leads to the existence of a polarization
wave that represents the response of the medium
to this perturbation. The polarization wave
(through the variable fields it generates) in
turn modifies the propagating wave. The actual
wave is the complicated result of these entangled
actions and retro - actions.
It also follows from this coupling that the role
of each wave on the energy transport is hard to
dissociate. Besides, the fields and charge
densities being spatially averaged quantities, it
is not obvious that the energy transfers between
wave and matter can be validly described by using
these averaged quantities. It can however be
shown that the energy flux through a closed
surface which is due to the propagation of the
e-m wave in matter is equal to the flux of the
Poynting vector R where 
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Propagation of an electromagnetic wave in a
dielectric medium The medium is linear, non
magnetic and isotropic
E, B, k form a trihedral
Dispersion relation a necessary compatibility
requirement
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The dielectric constant is a priori a complex
number. To satisfy the dispersion relation it is
therefore necessary that the wavevector comprises
a non vanishing imaginary part. Let er e1ie2
and let the wave to propagate along the x axis k
kex. By writing that k k1 ik2, the
dispersion relation leads to The
electrical field of the e-m is thus
written Equivalently, one can define
a complex refractive index
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The phase velocity of the wave is  By
averaging the Poynting vector over one period of
the e-m (because in general no detector is fast
enough to record oscillation at the 2? angular
frequency), we get This shows that the
energetic flux of the e-m wave decays
exponentially along the propagation direction.
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As one can realize easily, the progressive plane
waves that are solutions of the Maxwell equations
are not normalizable. Actually, plane waves are
everywhere for ever since whatever the spatial
location, it is always possible to find a time
that gives any prescribed value to the phase of a
plane wave.
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Then
 central  plane wave
Amplitude decays to zero at large r
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Conclusion The main conceptual difficulty is the
averaging procedure of the atomic distribution of
charges and fields in matter. This averaging is
to a large extent independent of the material
(insulator, metal, semiconductor) and is well
justified for a large part of the e-m spectrum.
To understand the wave propagation in linear
materials, we need to evaluate their frequency
dependent dielectric constant of the material.
Hence, we shall have to elaborate models for ????
starting by the simpler one, which is the Drude
Lorentz model.