RP Fiber Power – Simulation and Design Software for Fiber Optics, Amplifiers and Fiber Lasers

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Case Study: Mode Structure of a Step-index Multimode Fiber

We consider a step-index fiber with a core diameter of 20 μm and a numerical aperture of 0.15, to be used at a wavelength of 1060 nm. We want to analyze the mode structure not only for the nominal step-index design, but also for some types of modifications which typically can occur in practice.

For this case study, we use the Power Form “Mode Properties of a Fiber”.

Analysis of the Nominal Fiber Design

Let's start with the nominal design. It is simple to enter these few parameters into the form, execute it and see some outputs in the same form (see the fields with gray background):

Power Form for calculating fiber mode properties — Figure 1: Part of the Power Form for calculating mode properties.

So we quickly learn that we have 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 – for example, the group index – are missing here. This is because one would need more input data for those: the wavelength dependence of the refractive indices. For germanosilicate fibers, we have a Power Form where we can indirectly define the index profile via the GeO₂ concentration, thus getting the wavelength dependence and allowing for dispersion calculations.

You may wonder if 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: LP₃₃ with 75.9% and LP₁₄ with 47.0%. It turns out that these also have an effective refractive index very close to the cladding index (here assumed to be 1.44). We suspect that those are close to their cut-off wavelength. Indeed, when we increase the wavelength only by 6 nm to 1066 nm, LP₁₄ has disappeared, while we need to go up to 1082 nm to get rid of LP₃₃ as well.

Going back to 1060 nm and inspecting LP₁₄, we see how its power extended beyond the core:

Figure 2: Screen shot of a diagram window where we can inspect mode intensity profiles.

It is also interesting to look at the far field:

Figure 3: Far field intensity profile of the LP₁₄ mode.

In contrast to all other modes, a substantial part of the power is here concentrated in the central region, apart from two weak rings at larger radii. This is related to the large spatial extension of the mode.

Super-Gaussian Index Profile

Now we want to 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 constant, and we now have some index variation beyond 10 μm radius. The form now looks like that:

Figure 4: Definition of a super-Gaussian refractive index profile.

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

Figure 5: The super-Gaussian refractive index profile and the effective mode indices.

By comparing this with the same diagram for the step-index profile (not shown here), one sees that all effective mode indices have been reduced a bit, except that the top two (LP₀₁ and LP₁₁) stay nearly unchanged. The number of modes has been slightly reduced to 20, or 36 when counting different orientations.

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:

Figure 6: Index profile with a dip in the center.

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 LP₀₁ now has an effective mode area of 209.2 μm², while for the step-index case we had only 176.5 μm².

Conclusions

We can study in detail how the mode structure of a fiber changes when its refractive index profile is modified in some way.

You see that the calculation of fiber modes is straightforward with the RP Fiber Power software: simply enter its parameters, possibly involving an expression for the refractive index vs. radial position, and inspect the resulting numerical and graphical outputs. You can rapidly explore parameters and configurations. The Power Form gives you plenty of freedom, e.g. for defining complicated index profiles, besides offering various useful diagrams.