Hi Dr. Popoff,
I have been trying to use the pyMMF module shared on your website for simulation of multimode fiber modes and am concerned about the appearance of the simulated modes for graded index fibers.
I used your code in Example 1 that you provide in the readme file on github. Additionally, for a comparison I also computed the modes for the same parameters using their Laguerre-Gauss (LG) polynomial approximations. I show in the attachment below three examples of LP modes (real part) simulated using the pyMMF module (top row) and the LG polynomial approximation of the same modes (second row). I was expecting to see similar results, but some of the modes simulated using pyMMF appear in a line or square geometry. Is this normal? I was expecting both LP modes and LG modes to have circular geometries and similar profiles.
Could you please throw some light on the difference between the two representations?
Thanks!

Hi Sakshi,

I was expecting both LP modes and LG modes to have circular geometries and similar profiles.

That is where your problem is. You forgot that modes can be degenerate.

In a nutshell, if a mode is not degenerate, it sure will share the same symmetry as the system. But when you have a group of degenerate modes, the group should share the symmetry but not the modes individually. In the case of a multimode fiber, you could easily check that if you rotate by any arbitrary angle the modes in a given subspace (i.e. corresponding to the same propagation constant), they are still a basis of your subspace. However, each LP mode associated with a degenerate propagation constant is not circularly symmetric.

For instance, the two LP11 modes oriented horizontally and vertically are orthogonal modes with the same propagation constant. The same profiles oriented at + and -45 degrees are too.

If you play with the phase, you can find linear combinations of degenerate modes that look totally different from the ones you started with.

In the case of graded index fiber, it is interesting to notice that the size of the groups of degenerate modes increases when the propagation constant decreases. So that you can find more combination of modes.

I am not familiar with the LG modes for MMF, but it seems that they are a mode basis for graded index fiber but not for step index fibers.

To check if both representations are similar, simply project each LG modes on all the LP modes. If everything is right, for each LG mode, the only non-zero values should correspond to LP modes with the same propagation constant. This would show that the two representations are good.

Best,

Sebastien

Thanks for the detailed explanation. This makes sense and the test you suggested by projecting LG modes onto LP modes also checks out.