Chapter 11

Electromagnetic waves

Interfaces between two medium


In the matter, an electromagnetic field acts on the atoms, molecules or ions by deforming them. The material medium is as follows :

  • Polarize under the influence of the electric field.

  • Magnetize under the action of the magnetic field.

In some mediums (metal, plasmas), the charges move freely (we talk about free charges and currents).

In dielectric (or insulating) the charges do move only very little and are organized mainly in the form of electric dipoles (neglecting the contributions due to the apparition of quadrupole or even electric octopoles).

These changes will then cause the apparition of an electric field (or magnetic) induced by the external electromagnetic field, which is added thereto.

One mistake is to believe that an insulating medium (such as dielectric) does not let the waves pass through (radio for example).

This is exactly the opposite : the dielectrics are essentially transparent to radio waves, while conductor medium (sea water, metals) reflect and absorb them.


FondamentalSpeed of an electromagnetic wave in a dielectric

We define the index of a dielectric medium (such as glass, for example), in the form :

Where is the speed of light in vacuum and the velocity of light in the medium under consideration.

For glass, is about and for water, is about .

Vacuum has an index equal to and the air was very close to that of vacum index.

D'Alembert's equations for electromagnetic fields then become, replacing by  :

FondamentalReflection and transmission coefficients (in amplitude)

Reflection and refraction of electromagnetic waves : reflection and transmission coefficients (in amplitude) at normal incidence

Consider two dielectric mediums and .

It is located in the transparent areas for which the index of the two medium and are real.

An incident wave reaches the interface between the two medium (onto the interface surface between the two dielectric) at normal incidence.

We wish to determine reflected wave and transmitted wave.

Reflection and refraction of electromagnetic wave

The polarized waves is assumed linearly :

Note the wave vectors  ( is the wave vector in vacuum) :

Also :

  • Expressions of electric fields :

  • Expression of magnetic fields :

    One obtains the magnetic fields using the relation of the structure of a plane wave.

    So :

    Or, finally :

    Similarly (the reflected wave propagates in ) :

    And :

Dielectrics are uncharged. In a volumetric representation, there is continuity of the electric and magnetic fields at the crossing in the diopter at .

Therefore, the boundary conditions for the fields are written :

  • For the electric field :

  • For the magnetic field :

We can deduce :

is the amplitude reflection coefficient for the electric field.

And :

is the amplitude transmission coefficient for the electric field.

Some notes :

  • is always positive : there is no phase change during transmission.

  • can be positive or negative :

    : reflection introduces no phase shift

    : reflection introduces a phase shift equals to .

FondamentalReflection and transmission coefficients (in energy)

The incident Poynting vector is (on average) :

For reflected and transmitted waves :

The coefficients in energy of reflection and transmission wave are then :

We verify the conservation of energy :

A numerical application :

In the case of the diopter air-glass ( and ) :

and  : practically, energy is completely transmitted.

SimulationJAVA Animation (Jean-Jacques Rousseau (University of Le Mans))

  • Wave reflection : click HERE

  • Refraction of a plane transverse wave : click HERE

Electromagnetic waves in plasmas
Radiation pressure (corpuscular calculation)