RP Photonics logo
RP Photonics
Technical consulting services on lasers, nonlinear optics, fiber optics etc.
Profit from the knowledge and experience of a top expert!
Powerful simulation and design software.
Make computer models in order to get a comprehensive understanding of your devices!
Success comes from understanding – be it in science or in industrial development.
The famous Encyclopedia of Laser Physics and Technology – available online for free!
The ideal place for finding suppliers for many photonics products.
Advertisers: Make sure to have your products displayed here!
… combined with a great Buyer's Guide!
VLib part of the

Paraxial Approximation

<<<  |  >>>  |  Feedback

Buyer's Guide

Use the RP Photonics Buyer's Guide to find suppliers for photonics products! You will hardly find a more convenient resource.

Definition: a frequently used approximation, essentially assuming small angular deviations of the propagation directions from some beam axis

German: paraxiale Näherung

Category: general optics

How to cite the article; suggest additional literature

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 Geometric Optics

Geometric optics (ray optics) describes light propagation in the form of geometric rays. Here, the paraxial approximation means that the angle θ between such rays and some reference axis of the optical system always remains small, i.e. < < 1 rad. Within that approximation, it can be assumed that tan θ ≈ sin θ ≈ θ. 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 offset and angle of beams before and after some optical component or system.

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 (as obtained from Maxwell's equations) can be replaced with a simple first-order equation. 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 θ = λ / (πw0) 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.

See also: ABCD matrix, Gaussian beams, fibers

How do you rate this article?

Your general impression: don't know poor satisfactory good excellent
Technical quality: don't know poor satisfactory good excellent
Usefulness: don't know poor satisfactory good excellent
Readability: don't know poor satisfactory good excellent

Found any errors? Suggestions for improvements? Do you know a better web page on this topic?

Spam protection: (enter the value of 5 + 8 in this field!)

If you want a response, you may leave your e-mail address in the comments field, or directly send an e-mail.

If you like our website, you may also want to get our newsletters!

If you like this article, share it with your friends and colleagues, e.g. via social media:

Bragg mirror

Evolution of the reflectivity spectrum of a Bragg mirror during fabrication.

This diagram has been made with the RP Coating software.

– Show all banners –

– Get your own banner! –