# Simulating Non-monochromatic Multimode Beams

Posted on 2020-09-10 in the RP Photonics Software News (available as e-mail newsletter!)

Permanent link: https://www.rp-photonics.com/software_news_2020_09_10.html

Author: Dr. Rüdiger Paschotta, RP Photonics Consulting GmbH

Abstract: Laser beams with non-ideal beam quality can generally not be realistically simulated with numerical beam propagation based on a monochromatic field. However, with suitable software such as RP Fiber Power one can implement a multi-frequency technique which produces very realistic results.

In March 2020 I presented an article on creating multimode beams in numerical beam propagation. Here, I would like to add some important aspects to that topic. That came to my mind when developing a relatively complex simulation model for bulk lasers and amplifiers. The reasoning involves some non-trivial technical aspects, but may be of interest (a) for those developing such models and (b) also for others just wanting to understand more precisely how laser beams behave.

The pump and signal input beams of bulk lasers and amplifiers are not always diffraction-limited. This raises the question how to properly treat that in a simulation model. My first thought was to effectively assume monochromatic input beams, which would be somewhat “spoiled” in order to achieve the wanted degree of beam quality – using some of the methods described in my previous article. However, is that a physically realistic approach?

In short, the answer is that it depends on how we create such a non-ideal input beam – and usually it is *not* realistic.
It could be a realistic if in reality we would start with a laser beam having perfect beam quality, and then spoil this by sending the beam through some distorting medium (e.g. a somewhat inhomogeneous glass plate).
However, the usual situation is that we have a bulk laser, the beam quality of which is not perfect due to the fact that some (or even most) of its output power is in higher-order transverse modes.
Note that those modes usually have optical frequencies which somewhat deviate from those of the fundamental (Gaussian) modes.
(This is because of the mode-dependent Gouy phase shift.)
As a result, the output of such a laser is necessarily polychromatic.

## Does Monochromaticity Matter for Beam Propagation?

Well, at first glance one may think that this does not matter, since the overall spectral width of the laser output may be so small that we can effectively consider the beam as quasi-monochromatic. To all, the involved wavelengths are often so close to each other that things like beam divergence do not significantly vary within the optical spectrum. However, with that we would overlook the fact that due to the mode-dependent frequencies we have no interference effects between different modes – at least on an observation timescale which is too long to resolve fast optical beat notes.

Therefore, it is not realistic to simulate the propagation of such a non-ideal beam simply with numerical beam propagation based on a complex amplitude grid which resembles a monochromatic field. That approach would deliver artificial interference effects which are not observed in reality. By looking at beam profiles, one can easily see complicated interference patterns with a strong intensity modulation (see Figure 1), and those are clearly not there in most real multimode laser beams.

## How to Do It Correctly

Therefore, I decided to use the following concept:

- For a particular input beam, we decide for some maximum mode index, and in the simulation we separately propagate all TEM
_{n,m}modes with*n*and*m*up to that value. - The question is then how to distribute the total optical power over those modes.
For that, we can use a simple formula – for example, a supergaussian function on the mode indices, providing highest power for the fundamental mode and gradually reduced powers for the higher-order modes.
We can adjust the parameters of that function such that we obtain the wanted beam quality
*M*^{2}factor. - The optical phases of those modes do not matter, since we assume that different transverse modes do not interfere with each other – we therefore simply add up the resulting optical intensities to obtain the overall beam intensity. This approach may not be perfectly realistic because we ignore possible mode frequency degeneracies, but these will in practice usually not play a significant role – for example, when such degeneracies are lifted by slight asymmetries in the optical setup.

Figure 2 shows an example result calculated with such a model within a couple of seconds. It does not exhibit the very strong spatial intensity modulations of a monochromatic and therefore not realistic model.

The explained strategy is not that is difficult to implement e.g. with our powerful software RP Fiber Power, which we can control with a script language. (Using a software without scripting support, it would be hard to impossible, of course, unless just such a kind of model is already included in hard-wired form.) The only downside is that in cases with many modes we need to perform correspondingly many separate mode propagations, and that leads to longer computation times. Well, that is actually not that bad, since I recently introduced the Numerical Power Package which improves the computation speed in such cases by roughly an order of magnitude. With that, the simulation of pumping an amplifier and then amplifying a nanosecond pulse can be performed on an ordinary PC within a couple of seconds, as long as the number of modes is relatively small. Even in highly multimode situations, we would be far from running the computer over night.

Of course, overall such things tend to be somewhat tricky to implement, and most users of our software may find it too tricky to be practical – assuming that they would have to develop it themselves. However, within the technical support I can of course provide diligent help – for example, delivering example scripts for implementing such strategies – apart from giving general advice on what simulation strategy is appropriate in a certain case.

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