# Case Study: The Numerical Aperture of a Fiber: a Strict Limit for the Acceptance Angle?

Key questions:

- Why can the angular distributions of fiber modes go somewhat beyond the limit according to the numerical aperture?
- How is that situation for single-mode, few-mode and highly multimode fibers?
- How about modes close to their cut-off?

## Analytical Considerations

Common wisdom is that for guiding light in an optical fiber we need to fulfill the criterion for total internal reflection. That leads to a limitation of the incidence angle (acceptance angle) <$\theta$> of a light ray according to

$$\sin \theta \leq {\rm NA} = \sqrt{n_{\rm co}^2 - n_{\rm cl}^2}$$where NA is the numerical aperture (NA) of the fiber, which is calculated from the refractive indices of fiber core and cladding. For simplicity, here we consider only step-index fibers.

Note that angular distributions, obtained by spatial Fourier transformation of the mode amplitudes, are strongly related to the far field of light emerging from a fiber end. At a sufficiently large distance, the intensity profile of that light in a plane transverse to the propagation direction is determined by the spatial Fourier spectrum of the field amplitudes (at least within the paraxial approximation). (See the article on Fourier optics for more details.)

It is known that the guided modes of a fiber have far field intensity distributions which do not abruptly vanish at a certain boundary. This raises the interesting question whether the limit for the incidence angle, which is directly connected with the transverse component of the wave vector, should actually also lead to a limit of that angular intensity distribution. If not, wouldn't that imply that the modes become lossy, as total internal reflection does not work for some part of their angular spectrum?

To understand that, we need to consider that the acceptance angle according to the numerical aperture is based on a concept which mixes two rather different kinds of theories:

- It considers total internal reflection, a concept of wave optics, which usually involves plane waves.
- The obtained angle limit is then applied to a light ray, which is a concept of geometrical optics.

Any light which is guided in a fiber core can, of course, not be a plane wave, as it is by definition spatially limited to the fiber core (possibly including its immediate vicinity). That spatial confinement naturally leads to effects of diffraction, thus to a continuous angular distribution. This is fundamentally different from the situation of a plane wave, which is not transversely limited and does not exhibit diffraction.

From these considerations, we can see that a strict angular limitation does *not* apply to fibers, as this is possible only for plane waves, which we do not have here. And the mode calculations result in amplitude distributions which do *not* exhibit propagation losses due to non-total internal reflection.

Still, it is interesting to investigate under what circumstances and to which extent the angular distributions of light in fibers (or emerging from fiber ends) can exceed the NA limit. That we will do in the following.

## Numerical Tests

We address the question with some numerical tests, using the RP Fiber Power software. That offers a Power Form titled “Fiber Modes From Refractive Index Profile”, which is ideally suited for such investigations. In the following, we do our tests in several different situations. For all those, we assume the same fixed value 0.2 of the numerical aperture and a wavelength of 1 μm, so that we can directly compare the results.

### Single-mode Fiber

For a single-mode step-index fiber, there is one parameter left to be determined, which can have a significant effect on the results: the core radius. We can expect the highest beam divergence of the fiber's output light when the mode radius is smallest, and the mode radius depends on the core radius:

- If we use a rather small core radius of 0.5 μm, the mode radius is very large (63.2 μm). The mode reaches far beyond the core, and that leads to a very weak beam divergence, by far not reaching the limit given by the NA. Such a mode is very weakly guided and would be very sensitive to bending of the fiber, for example. Therefore, this is not a typical case.
- Note that here we have a situation where light approaching the limit according to the NA can not be guided – despite the fulfilled condition for total internal reflection!
- With a core radius of ≈1.62 μm, we get the smallest possible beam radius of 2.06 μm. Here, we have 74.6% of the optical power within the core. The far field extends significantly beyond the limit based on the NA, indicated by the gray vertical line:

6.3% of the optical power are outside the limit based on the NA. For still larger cores, resulting in a larger LP_{01} mode radius, we again have less power outside that limit.

You may wonder how that works out for different values of the NA. Instead of trying this numerically, we can use some simple equations:

- Let us assume that we have a V number of 2, which is reasonable for a single-mode fiber, and is quite accurately fulfilled in our case. From that, we can calculate the core radius:

- For that V number, we know that the mode radius is always a bit larger than the core radius – approximately by a factor of 1.3.
- From that, we can calculate the beam divergence half angle, assuming a Gaussian mode shape (which is approximately given):

From this, we can conclude that the divergence will always be somewhat below the NA limit, but a few percent of the angular spectrum can still be outside that limit. For other V values, it will be less.

So far, this case we would actually not have required a numerical test. In the following, however, we consider cases where a numerical tool is really useful.

### Few-mode Fiber

Next we consider a fiber with a somewhat larger core radius of 5 μm, where we have 6 modes (10 when counting all orientations):

The mode radius of the fundamental mode (LP_{01}) is now significantly larger (3.88 μm), which reduces its divergence; its far field is now fully within the NA limit.

That is not the case of some of the higher-order modes, namely for LP_{31} and LP_{21}, which go significantly beyond the NA limit. (For the meaning of those symbols, see the article on LP modes.)

Interestingly, not all modes close to their cut-off wavelength show that behavior. For example, if we increase the core radius to 5.75 μm, we get the LP_{03} close to its cut-off wavelength of 1030 nm. Here, the mode field extends substantially beyond the core, but the far field distribution is rather narrow, thus mostly within the NA limit (including the faint ring around the center structure):

For comparison, take the far field profile of the LP_{22} mode, which is more typical:

### Highly Multimode Fiber

Finally, we increase the core radius to 25 μm, were the number of modes rises to 129 (or 248 when counting all orientations). In this case, only some of the highest-order modes extend substantially beyond the NA limit:

For such fibers, the excitation of specific modes is normally not realistic; rather, we get some not very controlled distribution of optical powers over all the modes. Let us for simplicity assume that each mode gets the same power, and we consider an incoherent superposition of their intensity profiles:

Here, we see a quite steep drop at the NA limit (again indicated with a vertical line), and not much power outside that limit.

In practice, one will often launch little power into the highest-order modes, and the part extending the NA limit will then be smaller.

## Conclusions

We have learned various things from this study:

- The simple rule for total internal reflection, which is based on plane waves, cannot be fully applied to the situation of a fiber. Therefore, it is physically possible for the far field distributions to somewhat extend beyond the NA limit.
- On the other hand, there are some special cases of single-mode fibers where the fiber mode does by far not utilize the full angular range as allowed by the NA.
- In multimode fibers, typically only some of the highest-order modes extend beyond the NA limit, but even some of those (close to their cut-off) stay well within it.

A combination of analytical considerations and numerical tests give one the best chances to fully answer such questions.

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