Paraxial wave equation and spherical wave.

Any electromagnetic wave propagating inside an homogeneous medium verifies the Maxwell's equations. A direct consequence is that in an isotropic medium the propagation equation is as follows :

If we consider the propagation of a monochromatic electromagnetic radiation with a frequency , we can write this equation in a different manner and show that the wave must verify the Helmholtz equation :

where  is the wavevector.

This equation have a very well-known solution : the diverging spherical wave, which can be written :

where the source is located at (x,y,z) = (0,0,0) and r is the distance from the origin.

In the paraxial approximation framework, we assume that the wave propagation is along a specific axis (z-axis). In this case, we can use the following Taylor development :

The electric field for the position r is then :

It represents the field for a “paraxial spherical wave”, which is only an approximate solution of the Helmholtz equation. We can recognize the propagation factor exp{-ikz} as well as the transverse variation of the amplitude :

From a mathematical point of view, the spherical wave is a solution of the propagation equation. From a physical point of view, the paraxial spherical wave is an acceptable approximate solution to describe the wave propagation.

However, in our case (that is, for lasers), this wave is not a convenient solution : the energy spreads out in all directions, and when we isolate the paraxial part a great amount of energy is lost, which is not compatible to efficient laser operation. Indeed, the electromagnetic field structure inside an optical resonator should ideally verify the following conditions :

• Verify the Maxwell's equations

• The field amplitude should decrease when the distance relative to the cavity axis increase, to take into consideration the finite dimensions of the mirrors and of the gain medium.

• The wavefront must fit to the radius of curvature of the mirrors (this condition exclude plane waves)

We will now describe the solutions that are well-adapted to laser resonators.