Case Study: Number of Modes of a Highly Multimode Fiber

Key questions:

What determines the number of guides modes of a multimode fiber?
Can we generalize a well-known equation (based on the V number) which holds only for step-index fibers, so that we have an estimate for arbitrary index profiles?

For multimode fibers, it can be of interest how many guided modes they support. For example, a fiber may support a moderate number of modes like those:

Figure 1: The modes of a fiber for a certain wavelength.

Some multimode fibers have many more modes, e.g. several thousands. Here is an example with nearly 1000:

example with many fiber modes — Figure 2: Nearly 1000 modes of a fiber with larger <$V$> number. As the space is insufficient for showing all mode profiles accurately, we show effective refractive index values instead.

For fibers with a simple step-index design (i.e., with constant refractive index inside the fiber core) and sufficiently high V number, it is known that the number of modes can be estimated as <$V^2 / 4$> per polarization direction, or twice that number for both directions. The central question of this article:

Can we generalize that rule to fibers with arbitrary index profiles?

Note that here we always count different orientations for <$l \neq 0$> separately; for example, a fiber supporting only LP₀₁ (fundamental mode) and LP₁₁ would be considered as having three modes, since LP₁₁ can have two orientations (with a <$\sin \varphi$> or <$\cos \varphi$> azimuthal dependence). The fiber according to Fig. 1 has 6 modes in that sense, and the one of Fig. 2 has 994.

Developing a Hypothesis

We find a suitable equation with a heuristic approach. We first consider the V number, which is defined as

$$V = \frac{{2\pi }}{\lambda } r_{\rm co} \;{\rm{NA}}$$

with the core radius <$r_{\rm co}$>, the numerical aperture NA and vacuum wavelength <$\lambda$>. With the known equation for an estimate, this results in the following formula for the approximate number of modes (for one polarization direction):

$$M \approx V^2/4 = \frac{\pi}{\lambda^2} \; \pi \; r_{\rm co}^2 \: {\rm NA}^2 $$

We recognize <$\pi \; r_{\rm co}^2$> as the core area, while <$\pi$> NA² determines the solid angle defined by the numerical aperture (within the paraxial approximation). So it is reasonable to guess the following generalized formula:

$$M \approx \frac{\pi}{\lambda^2} A_{\rm co} \: \left< n_{\rm{co}}^2 - n_{\rm{cl}}^2 \right> $$

where the angle brackets in <$\left< n_{\rm{co}}^2 - n_{\rm{cl}}^2 \right>$> mean averaging over the whole core area (kind of the “averaged NA squared”).

Note that assuming a non-magnetic medium, as usual in optics, we have <$n^2 = \epsilon_{\rm r}$>, the dielectric constant, so that effectively we average simply the increase in dielectric constant. Also, we can write <$n_{\rm co} = n_{\rm cl} + \delta n$>, from which we see that the angle bracket becomes approximately <$2 n_{\rm cl} \left< \delta n \right>$> (neglecting the small term with <$(\delta n)^2$>).

We can also write this equation with a radial integral:

$$M \approx \frac{\pi}{\lambda^2} \int_0^{r_{\rm co}}{2\pi \; r \: \left(n^2(r) - n_{\rm{cl}}^2\right) \: {\rm d}r}$$

$$M \approx \frac{\pi}{\lambda^2} \int {\left(n^2 - n_{\rm{cl}}^2\right) \: {\rm d}A}$$

(again with an integration over the whole core area).

This can be interpreted as a kind of volume under a 3D plot of <$n^2(r) - n_{\rm{cl}}^2$> over the core area, multiplied with the factor <$\pi / \lambda^2$>.

Proving that rule mathematically might well be possible (does a reader know the proof?), but probably not so easy. Instead, we just subject our hypothesis to some other type of tests:

Numerical Tests

We conveniently use the software RP Fiber Power, which offers the Power Form “Mode Properties of a Fiber”.

Step-index Fiber

We start with a step-index fiber with 100 μm core diameter and an NA of 0.2, for which the software calculates 994 modes, while the estimate <$V^2 / 4$> is 987 modes. We let the form display the ratio of the calculated to the expected number of modes, which is 1.01 in this case.

Ring Profile

The next test is with an increased-index ring (with the same maximum index as before) ranging from <$r_{\rm co} / 2$> to <$r_{\rm co}$>. Here, the number of modes drops to 747, while the estimate drops by one quarter to 741. Our ratio is still 1.01 – fine.

Figure 3: Ring index profile.

That also works when the ring only extends from 80% to 100% of the core radius: 363 modes calculated, 356 expected. The ratio increases to 1.02; that's just because the estimate works less well as the number of modes drops.

Parabolic Profile

We consider a parabolic profile, having the same index increase as for the step-index fiber in the center, but with a parabolic decay.

Figure 4: Parabolic index profile.

That reduces the number of modes to 496, again 1.01 the estimate.

Inverse Parabolic Profile

Now we have the cladding index in the center, with a parabolic increase up to the cladding, where the index suddenly drops to the cladding index:

Figure 5: Inverse parabolic index profile.

That may not be a realistic index profile, but our estimate works again: 500 modes = 1.01 the estimate of 493.

Wavelength Dependence

Our formula shows that the wavelength dependence of the number of modes is always the same: an inverse quadratic one – no matter what is the shape of the index profile. We can also test this numerically. For example, the parabolic profile case mentioned above can be tested for 500 nm wavelength instead of 1000. The number of modes rises from 496 to 1960, i.e., by a factor 3.95, close to 2² = 4.

By the way, the inverse square dependence is related to the fact that light in fiber in confined in two dimensions. For a one-dimensional waveguide, we would only have a <$\lambda^{-1}$> dependence.

Conclusions

Our tests all confirmed the hypothesis that we can well estimate the number of guided modes of a fiber by considering the average squared refractive index (relative to the squared cladding index) over the core area. While this is not a fail-safe proof, these tests combined with physical intuition give us considerable confidence.

Note that we could do the tests only for fiber designs with a radially symmetric profile as the LP mode solver works only in that case. But it is plausible to assume that the formula works even for not radially symmetric cases.

By the way, I asked ChatGPT and Google Gemini, two common artificial intelligence tools, whether one could generalize that equation; both failed. If you are interested in AI, there is an article on the performance of AI in photonics.

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