# Case Study: Mode Structure of Multimode Fibers

Key questions:

- Are the mode profiles all strongly confined to the fiber core?
- What happens for modes close to their cut-off?
- Do higher-order modes have larger effective mode areas?
- Why do the effective refractive indices vary so much?
- How to calculate wavelength-dependent mode parameters?
- What changes if the index profile is somewhat smoothened, or when it obtains a dip at the center?

Here, we investigate various interesting features of the guided modes of multimode fibers. By thoroughly looking at those, one can learn a lot on fiber optics.

For this case study, we use the software RP Fiber Power – initially, with its Power Form “Mode Properties of a Fiber”.

## Analysis of a Step-index Fiber

We consider a step-index fiber with a core diameter of 20 μm and a numerical aperture of 0.2, to be used at a wavelength of 1060 nm.

Initially, we directly specify the refractive index profile, initially choosing the “single step” variant, which is the simplest for a step-index design. For this, we need only few parameters: the core diameter, the cladding index and the numerical aperture. By executing the calculation, we get various output quantities displayed (fields with a gray background), namely the index of the core and the spanned refractive index range (here, simply the indices of cladding and core).

The fiber supports 21 different guided modes, or 38 when separately counting the different orientations. (All LP modes with <$l \neq 0$> exist in two different orientations.) The table shows the essential properties of all those modes: <$\beta$> = propagation constant (phase constant), <$n_{\rm eff}$> = effective refractive index, <$A_{\rm eff}$> = effective mode area, <$P_{\rm core}$> = fraction of the optical power which propagates in the core.

Note that quantities related to chromatic dispersion, and also the group index, are missing here. This is because one would need more input data for those: the wavelength (frequency) dependence of the refractive indices. The same holds for the cut-off wavelength of each mode. We will later use another Power Form for germanosilicate fibers, where we can indirectly define the index profile via the GeO_{2} concentration, thus getting the wavelength dependence and allowing for dispersion calculations. But we first consider various properties of the modes as calculated for one given wavelength.

The radial mode functions are often of interest:

A perhaps more useful overview on the modes is given by a diagram for the mode amplitude profiles:

Here, you can easily find the profile of the LP_{32} mode, for example: the diagram axes indicate the mode indices.

For shorter wavelengths, we would get more guided modes. For large values of the V number, which is inversely proportional to the wavelength, the number of modes is roughly <$V^2 / 2$>, when counting both polarization directions. See the article on multimode fibers for more details.

### Effective Mode Areas

It is often believed that higher-order modes have significantly larger effective mode areas, as their profiles reach out further in the radial direction. Interestingly, however, our calculations do not confirm that for this quite common example case. In fact, the mode area of LP_{01} (176.5 μm^{2}) is even a bit larger than that of all other modes, except only for LP_{14} with 343.2 μm^{2}. For example, LP_{42} has only 153.6 μm^{2}. Let us inspect their radial intensity profiles:

You see that while the LP_{01} mode covers a substantial portion of the core area, the LP_{42} mode is more concentrated in three rings, and in addition it has the azimuthal variation of intensity.

For comparison, see the LP_{14} mode, really having a substantially larger effective mode area, and also extending far into the cladding, as it is close to its cut-off:

There are other cases where higher-order modes really have substantially larger mode areas, but here it is obviously not the case.

### Observations on the Effective Refractive Indices

For each mode, the form displays its effective refractive index. This is directly related to the propagation constant (phase constant) <$\beta$> displayed in the previous column: we have

$$\beta = n_{\rm eff} \frac{2\pi}{\lambda},$$which is analogous to

$$k = n \frac{2\pi}{\lambda}$$for plane waves. We can also display the effective indices of all modes together with the refractive index profile:

Another way to get a good overview is a color diagram:

The lowest order modes have the highest effective index values, close to the core refractive index, while the high-order modes have lower values, eventually approaching the cladding index.

One may think that the effective refractive index is reduced for higher-order modes because those have a larger fraction of their optical power propagating in the cladding; behind this is the idea that the effective index is a kind of weighted average of refractive indices. That is not true, however. Most of the modes have their power nearly entirely within the core:

In order to understand this, we just need to look at the previous equation. It shows us that the effective refractive index is related to the phase constant, i.e., to the phase change per unit length of propagation. The reduced values for higher-order modes indicate that those modes involve smaller phase changes during propagation. That is actually another effect which may appear quite surprising, as higher-order modes are often associated with rays traveling with substantial angles of inclination against the fiber axis. Shouldn't they exhibit *larger* phase changes due to effectively longer propagation paths? The answer is no: what is relevant, is the phase change in <$z$> direction, and that is *reduced* if the <$k$> vector is inclined, which implies a reduced <$k_z$> component. Note, however, that the relation between rays (a concept of geometrical optics) and modes (a concept of wave optics) is a complicated one.

### Are the intensity profiles constrained to the fiber core?

You may wonder whether the intensity profiles of all modes are more or less constrained to the fiber core. For that, we can simply inspect the last column of the table in the form. Most modes have well over 90% of their power within the core, but at the bottom (seen only after scrolling) there are two outliers: LP_{33} with 75.9% and LP_{14} with 47.0%. It turns out that these also have an effective refractive index very close to the cladding index (here assumed to be 1.45). We suspect that those are close to their cut-off wavelength. Indeed, when we increase the wavelength only by 6 nm to 1066 nm, LP_{14} has disappeared, while we need to go up to 1082 nm to get rid of LP_{33} as well.

Let us inspect the LP_{14} mode (for 1060 nm) more closely. The amplitude profile is shown in Fig. 3, but we can also plot its intensity profile separately to see more details:

We can see how the optical power extends well beyond the fiber core (with its outer boundary indicated as a gray circle) – more than for the other modes.

It is also interesting to look at the far field of that mode:

In contrast to all other modes (including some with a relatively high order), a substantial part of the power is here concentrated in the central region, apart from two weak rings at larger radii. One may have expected a large divergence for such a higher-order mode, but the narrow feature is related to the large spatial extension of the mode. For comparison, see the far field intensity profile of the LP_{33} mode, which is not so close to its cut-off and does not extend that far into the cladding:

So we see that most of the modes do not extend much beyond the fiber core; this is the case for only very few modes, which happen to be close to their cut-off. However, for any choice of wavelength, it is not unlikely to have that situation at least for one or a few modes, as the number of modes changes at many different wavelengths:

## Super-Gaussian Index Profile

Now we check how the mode structure changes when the edges of the step-index profile are somewhat rounded – for example, resulting from diffusion during fabrication of the fiber preform. For this, we now use a super-Gaussian profile with high order, approaching a step-index profile but with rounded edges. For this, we choose “expression” for the type of index profile definition and enter the super-Gaussian function for the refractive index vs. radial position. We also need to slightly increase the core diameter to 21 μm, because the software defines that as the radius outside which the refractive index stays exactly constant, and we now have some index variation beyond 10 μm radius. The form now looks like that:

By comparing this with the same diagrams for the step-index profile, one sees that all effective mode indices have been reduced a bit, except that the top two (LP_{01} and LP_{11}) stay nearly unchanged. The number of modes has been slightly reduced to 20, or 36 when counting different orientations.

Here are the slightly modified mode amplitude profiles:

We can also check the resulting profile and the effective mode indices:

Of course, we could easily test this with other mathematical expressions for smoothening the index drop. We could also insert a few lines of script code into the form for reading tabulated index profile data from a file with basically arbitrary format.

## Index Profile with a Dip

Our next test case is a refractive index profile with a dip in the center. This is actually more common for single-mode fibers, but we also try it here. We use the expression `1.453823 - 0.008 * gauss(r / (1 um))`

with a Gaussian dip. The result:

We get the same number of modes as for the super-Gaussian profile, but significant changes particularly for the lowest order modes. For example, the fundamental mode LP_{01} now has an effective mode area of 209.2 μm^{2}, while for the step-index case we had only 176.5 μm^{2}.

## Calculating Wavelength-dependent Quantities and Chromatic Dispersion Properties

Now we also want to investigate chromatic dispersion properties. With the previously chosen approach of directly defining the refractive index profile, this is not possible: we need wavelength-dependent refractive index values! But entering those into a form would be inconvenient.

For the usual germanosilicate fibers, we have a convenient solution: as the original input, we specify the GeO_{2} doping profile, from which the refractive index profile can then be calculated. This can be done using the known Sellmeier data of fused silica (SiO_{2}) and germania (GeO_{2}), and a formula which allows us to interpolate between those. As that works for any wavelength, we have the required inputs for calculating the dispersion properties.

There is actually a caveat: Sellmeier equations often deliver quite wrong results outside the wavelength range for which they were made, and that can, in particular, mess up calculations of cut-off wavelengths, which may be relatively long. A quick fix is to limit the wavelength argument of those index functions to 3 μm, for example, avoiding the peculiar behavior at long wavelengths. It does not really matter that the long-wavelength data are still not real, as this anyway in the strongly absorbing spectral region. We should just not take large values of cut-off wavelengths too seriously.

This approach has been implemented in our Power Form “Modes of a Germanosilicate Fiber”, which we use in the following. We go back to the step-index fiber design. For a given GeO_{2} doping concentration in the fiber core, the form displays (among many other properties) the numerical aperture. We quickly find out that 9.2% doping is required for the wanted NA of 0.2 at 1060 nm. Here are the results for that:

Note that we could now enter any other wavelength and get the modes calculated, always using the actual refractive index values. Also, we can make diagrams for wavelength-dependent quantities, such as the effective mode area:

The new table of mode properties contains three more quantities than before, which can be calculated only if we have the full wavelength dependence of the refractive indices:

- the group index
`n_g`

- the group velocity dispersion (GVD)
- the cut-off wavelength (not for the LP
_{01}mode, which does not have a cut-off)

We also inspect the effective indices and the group indices:

We have already seen before that the effective indices are related to the phase change per unit length, and that they generally get smaller for higher-order modes. The diagram also shows the group indices for plane waves in the core and cladding material with dashed lines.

For the group indices, the dependence on the mode index is different: here, we often get *higher* values for higher-order modes, indicating that the light pulses in those modes propagate with *lower* velocity. However, some modes exhibit a falling group index when approaching their cut-off. This effect gets more pronounced for modes with high `m`

indices.

## Conclusions

The RP Fiber Power software is an invaluable tool for such work – very powerful and at the same time pretty easy to use!

You can learn various things from this study:

- A multimode fiber with not too small core has plenty of modes, differing a lot in various respects.
- The effective mode areas of higher-order modes are not necessarily larger than those of the fundamental mode.
- Effective refractive indices should
*not*be interpreted as a kind of averaged index values. Instead, they are related to the phase properties, and get lower for higher-order modes. - Some mode profiles extend significantly into the fiber cladding. That is particularly the case for modes which are close to their cut-off. Curiously, in such cases one may have a relatively narrow main feature of the far field pattern.
- For investigating wavelength-dependent quantities and chromatic dispersion, we need wavelength-dependent refractive indices. Those can be obtained from Sellmeier equations.
- With a numerical model, one can investigate how exactly the mode structure reacts to changes of the design.

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