Helmholtz Equation - Paraxial Approximation

Paraxial Approximation

The paraxial approximation of the Helmholtz equation is:

where is the transverse part of the Laplacian.

This equation has important applications in the science of optics, where it provides solutions that describe the propagation of electromagnetic waves (light) in the form of either paraboloidal waves or Gaussian beams. Most lasers emit beams that take this form.

In the paraxial approximation, the complex magnitude of the electric field E becomes

where A represents the complex-valued amplitude of the electric field, which modulates the sinusoidal plane wave represented by the exponential factor.

The paraxial approximation places certain upper limits on the variation of the amplitude function A with respect to longitudinal distance z. Specifically:

and

These conditions are equivalent to saying that the angle θ between the wave vector k and the optical axis z must be small enough so that

The paraxial form of the Helmholtz equation is found by substituting the above-stated complex magnitude of the electric field into the general form of the Helmholtz equation as follows.

Expansion and cancellation yields the following:

Because of the paraxial inequalities stated above, the ∂2A/∂z2 factor is neglected in comparison with the ∂A/∂z factor. The yields the Paraxial Helmholtz equation.