Paraxial Approximation

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

What is the Paraxial Approximation?

Many calculations in [optics] can be greatly simplified by making the paraxial approximation, i.e. by assuming that the propagation direction of light (e.g. in some laser beam) deviates only slightly from some beam axis.

Paraxial Approximation in Geometrical Optics

Geometrical optics (ray optics) describes light propagation in the form of geometric rays. Here, the paraxial approximation means that the angle ($\theta$) between such rays and some reference axis of the optical system always remains small, i.e. much smaller than 1 rad. Within that approximation, it can be assumed that ($\tan \theta \approx \sin \theta \approx \theta$). The evolution of beam offset (distance from the reference axis) and beam angle in some optical system can then be described with simple ABCD matrices because there are linear relations between the offset and angle of a ray before and after some optical component or system. The paraxial approximation is extensively used in Gaussian optics .

Paraxial Approximation in Wave Optics

When describing light as a wave phenomenon, the local propagation direction of the energy can be identified with a direction normal to the wavefronts (except in situations with spatial walk-off). If the paraxial approximation holds, i.e. these propagation directions are all close to some reference axis, a second-order differential equation with respect to the ($z$) coordinate (as obtained from Maxwell's equations) can be replaced with a simpler equation which is of first order in ($z$). Based on this equation, the formalism of Gaussian beams can be derived, which gives a much simplified understanding of beam propagation and of fundamental limitations such as the minimum beam parameter product. Essentially, the paraxial approximation remains valid as long as divergence angles remain well below 1 rad. This also implies that the beam radius at a beam waist must be much larger than the wavelength.

The propagation modes of waveguides, particularly of optical fibers, are also often investigated based on the paraxial approximation. The validity of the analysis is then restricted to cases with a sufficiently large effective mode area and sufficiently small divergence of any beams exiting such a waveguide.

The paraxial approximation is very well fulfilled in a wide range of phenomena of laser physics and fiber optics, but it is clearly violated in cases with very strong focusing, where commonly used equations such as ($\theta = \lambda / (\pi w_0)$) for the divergence angle break down. In that regime, polarization issues also demand special care. In particular, polarization components in the propagation direction can occur. For such reasons, the simulation of beam propagation then requires significantly more sophisticated methods. For example, beam propagation methods (propagating a two-dimensional array of complex field amplitudes) can be used which do not need that approximation.

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