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 :

Δ E ⃗ − 1 c 2 ∂ 2 E ⃗ ∂ t 2 = 0 %DELTA vec E - 1 over c^2 {partial^2 vec E} over {partial t^2}=0

If we consider the propagation of a monochromatic electromagnetic radiation with a frequency \(\nu =\omega /2 \pi\), we can write this equation in a different manner and show that the wave must verify the Helmholtz equation :

Δ E ( x , y , z ) + k 2 E ( x , y , z ) = 0 %DELTA E (x,y,z) + k^2 E(x,y,z) = 0

where \(k=\omega /c\) is the wavevector.

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

E ( x , y , z ) = E 0 r exp ( − i k r ) E(x,y,z) = E sub 0 over r exp (-i k r)

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 :

r = x 2 + y 2 + z 2 ≈ z + x 2 + y 2 2 z r = sqrt{x^2 + y^2 + z^2} approx z + {x^2+y^2} over {2z}

The electric field for the position \(r\) is then :

E paraxial ( x , y , z ) = E 0 z e ( − i k z ) e ( − i k x 2 + y 2 2 z ) E sub paraxial (x,y,z) = E sub 0 over z e^(-i k z) e^(-i k {x^2 + y^2} over {2z})

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 :

1 z e ( − i k x 2 + y 2 2 z ) 1 over z e^(-i k {x^2 + y^2} over {2z})

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.