Propagating Optical Fields in Fibers Based on Modes or with Numerical Beam Propagation

Obviously, the computational effort of the mode-based approach does not depend on the propagation distance. It is only that for long distances small errors in the phase constants of the modes become more critical. This is not just a numerical issue: any real fiber will exhibit some fluctuations of its parameters and therefore of its phase constants. Therefore, the precise outcome cannot be predicted any more for a large propagation distance.

For numerical beam propagation, propagation distances of many meters in a fiber can become problematic, since a large number of numerical steps is required. Particularly for small-core fibers with high numerical aperture, the numerical step size needs to be rather small (often only a few micrometers), and millions of steps as required for several meters may challenge your patience. There can be also a problem with computer memory, if you need to store the amplitude profiles for all intermediate steps. Frequently, however, at least the latter problem can be easily solved by storing the fields only on a relatively coarse numerical grid.

In many cases, we can safely ignore cladding modes of the fiber, assuming that their losses over the relevant propagation distance are very high. For the mode-based approach this is very nice, since we have to deal with a usually quite limited number of modes. Four numerical beam propagation, it means that we have to consider only a relatively small volume, not extending that much beyond the fiber core. One should then simply include some artificial loss in the cladding area in order to get light removed from there. Otherwise, it would be reflected at the outer boundaries of the numerical grid and would thus produce artifacts.

Sometimes, however, cladding modes are relevant. For the mode-based approach, this can be a serious problem: we have to calculate all the cladding modes as well, and that can dramatically increase the number of modes, possibly even making the whole approach no longer feasible.

For numerical beam propagation, we simply have to extend the numerical grid such that it covers the full cladding area and also extends somewhat into the coating (or air). The price to pay is simply that we are dealing with a larger grid, requiring more memory and computation time.

If a fiber is bent, that breaks its radial symmetry. As a result, simple LP mode solvers can no longer be used. There are more sophisticated numerical mode solvers, which not every software offers, and which are computationally more demanding.

For numerical beam propagation, bending is relatively simple to include. Its effect can be simulated as a modification of the refractive index profile with an addition which is proportional e.g. to the x coordinate. A proper choice of the proportionality constant can be used to take into account the effect of mechanical stress as well.

Some fibers have properties which change along the length. For example, there are tapered fibers, where the core diameter varies gradually. Other fibers contain things like fiber Bragg gratings, e.g. long-period gratings which do not cause reflections but mode coupling – for example, coupling light from a guided mode into cladding modes.

The mode-based approach is usually not usable in such cases; it is not even obvious how to sensibly define modes of such structures. Well, there are specialized software products with sophisticated algorithms which are still usable at least for some of those problems. Personally, I do not have experience with that.

Numerical beam propagation is most flexible. Here, it is no problem to make the refractive index not only dependent on a radial coordinate r, or more generally on x and y, but also on the longitudinal coordinate z. Therefore, it is easy to investigate light propagation in tapered fibers, for example, or even fibers with random fluctuations of the core diameter. It is also not a problem to include a fiber Bragg grating.

Particularly in the context of optical pulses, which can have substantial peak powers, fibers nonlinearities can become quite relevant. One might think that a mode-based approach is that no longer usable, but often it still is. For example, one can iteratively calculate the modified modes of a fiber at a high optical power level; I have done this in a demo file for our software.

On the other hand, numerical beam propagation methods are again quite flexible. It is not too difficult to integrate optical nonlinearities. I have done that not only for the Kerr nonlinearity, leading to self-phase modulation and cross-phase modulation, but even for stimulated Raman scattering.

Sometimes it is necessary to include the time dependence of optical fields. For the mode-based approach, this simply means that the mode amplitudes become time-dependent.

For numerical beam propagation, it is also possible to introduce time dependencies, but note that the computational effort and memory requirements may easily explode: everything is multiplied with the number of temporal steps. Therefore, this is often not practical, or at least rather time-consuming and memory-intensive.

Anyway, not all is lost when time dependencies become important. There are cases where one investigates the nonlinear propagation of a nanosecond optical pulse, for example, by doing the propagation for several power levels corresponding to different temporal locations in the pulse, and then combining the obtained results. If the pulse duration is much longer than the group delay difference within the pulse spectrum, that can be a quite practical and efficient method.