## Fiber Modes

In general, the intensity profile of light propagating in a fiber changes during propagation. It often even evolves in a rather complicated way. As an example, see what happens if we inject a Gaussian beam, which is tilted by 20° against the beam axis, into a fiber with 20 μm core radius and an NA of 0.3:

Figure 1: Evolution of the intensity in a multimode fiber, simulated with the RP Fiber Power software. A Gaussian beam with an angle of 20° against the beam axis is injected into the fiber.

One clearly sees the interference effects occurring when the beam reaches the core/cladding interface and is reflected there. At the end, the transverse beam profile looks as shown in Figure 2:

Figure 2: Beam profile in the fiber after propagation over 100 μm.

We have seen the intensity profiles generally evolve in complicated ways. However, there are certain amplitude profiles (i.e., distributions of electric field amplitude) where the intensity profile remains unchanged during propagation (assuming a lossless fiber). Such field distributions are called *modes* of the fiber. The simplest of these, the *fundamental mode*, also called LP_{01} mode, looks as follows for the fiber in the current example:

Figure 3: Intensity profile of the fundamental mode in a multimode fiber. The gray circle indicates the core/cladding boundary.

As the cladding is typically much larger than the core, and also typically has a higher numerical aperture (due to a large index contrast to a coating around it), the cladding normally has many more modes than the core.

**Important Properties of Modes**

For a given wavelength, each fiber mode has several important characteristics:

- It has a certain complex amplitude profile, from which one obtains the intensity profile.
- From the intensity profile, one may calculate the effective mode area. This determines how strong nonlinear effects can be.
- The phase constant β tells how fast its overall complex phase changes during propagation. For strongly guided modes, it is well above the phase constant of the cladding. In that case, the intensity profile does not substantially extend into the cladding.
- From the first and second derivative of the β with respect to the optical frequency, one can calculate the group velocity and the group velocity dispersion (GVD).
- For lossy fibers, each mode can have a certain attenuation constant. These values can differ a lot between the modes – sometimes by orders of magnitude. For example, bend losses are often much higher for higher-order modes. They may be calculated with numerical beam propagation or estimated with some analytical techniques. By the way, bending can also shift and distort the mode profile.
- The cut-off wavelength is the wavelength above which the mode ceases to exist. (Under some circumstances, a mode can have a higher and lower cut-off.)

**Calculating the Propagation of Light Based on Modes**

Using the calculated set of mode, one can calculate the resulting field profile for an arbitrary input profile after any distance of propagation in the fiber:

- First, calculate the excitation amplitudes of all guided modes via complex overlap integrals of the input amplitude profile with the complex conjugates of all mode amplitude profile.
- Then change the complex amplitudes of all modes according to their β values.
- Construct the final beam profile by adding up all contributions from the modes.

Note that this procedure is not computationally difficult, unless the fiber has a huge number of modes and/or complicated modes in case of an index profile which is not radially symmetric. The amount of computations does *not* depend on the propagation distance. (For numerical beam propagation, longer distances would generally require more time.)

There are cases where mode coupling calculations are helpful. For example, one may calculate the modes of an “undisturbed” fiber and then calculate mode coupling caused by some additional effect. For example, a periodic index modulation in a fiber Bragg grating can couple counterpropagating or copropagating modes.

**Limitations of the Mode Approach**

The concept of modes is often very convenient for doing calculations, as shown above. However, it also has its limitations:

- In some cases – for example, for arbitrary index profiles – the modes are very difficult to compute.
- In cases with a huge number of modes, it may also not be convenient to do calculations based on them. Note: if cladding modes need to be taken into account, one would end up with a huge number of modes in the calculation, even if the core has only few guided modes.
- Additional disturbances such as bending already make the calculation of modes much more difficult.
- The concept is also at least more difficult to apply if the mode properties change along the length of a fiber – for example, for tapered fibers.

For such reasons, direct numerical beam propagation is often required instead of the mode concept.

## Single-mode Fibers

We have seen that depending on its refractive index profile and the wavelength, a fiber may guide different numbers of modes. It may be only a single guided mode (the LP_{01} mode), if the numerical aperture and the refractive index contrast are small. In this regime, the fiber is called a single-mode fiber. Higher-order modes like LP_{11}, LP_{20} etc. then do not exist – only cladding modes, which are not localized around the fiber core.

Note that in most cases light with different polarization states can be guided. The term “single-mode” ignores the fact that usually (for radially symmetric index profiles and no birefringence) one actually has two different modes with same intensity profile but orthogonal linear polarization directions. Any other polarization state can be considered as a linear superposition of these two.

**Condition for Single-mode Guidance**

For step-index fiber designs, there is a simple criterion for single-mode guidance: the V number has to be below ≈2.405. The V number is defined as

where λ is the vacuum wavelength, *a* is the radius of the fiber core, and NA is the numerical aperture.

For other radial dependencies of the refractive index, or even for non radially symmetric index profiles, the single-mode condition normally has to be calculated numerically. It would *not* be correct to use the criterion *V <* 2.405, e.g. with *V* calculated from the maximum index difference.

**Example: a Typical Single-mode Fiber**

A typical kind of single-mode fiber for 1.5 μm wavelength may have a step-index profile with a core radius of 4 μm and a numerical aperture of 0.12. The guided mode then has a mode radius of 5.1 μm and an effective mode area of 75 μm^{2}. That is not too far from the data of the often used SMF-28 telecom fiber from Corning.

Figure 4: Radial intensity profile of the LP_{01} mode of a single-mode fiber. The dotted curve shows a Gaussian profile, which is very similar. The gray vertical lines show the position of the core/cladding boundary.

As is typical for single-mode fibers, the field distribution extends significantly beyond the core; only 54.4% of the power propagates in the core. (It may seem according to Figure 4 that it is more, but note the factor *r* in the area integral, which makes the outer parts of the profile more important.) However, the intensity drops quickly with increasing radial coordinate. The intensity profile is close to a Gaussian profile.

When we decrease the wavelength, we find that the fiber is no more single-mode below the *cut-off wavelength* of 1254 nm: in addition to the LP_{01} mode, it then also supports LP_{11} modes (actually two of these with orthogonal orientation). Below 787 nm, the LP_{02} mode comes in additionally.

In principle, the fiber stays single-mode for any wavelength above the LP_{11} cut-off, which is 1254 nm. However, for longer wavelengths the mode becomes larger and larger, and it will become increasingly sensitive to bend losses, resulting both from macroscopic bending and from microscopic imperfections. For the design discussed here, another problem is actually more serious: beyond ≈2 μm, absorption of the base material (silica) sets in. So in practice there is a limited wavelength interval into which a single-mode fiber can be used.

**Launching Light into a Single-mode Fiber**

Efficiently launching light into a single fiber modes requires that the complex amplitude profile of the incident light (assuming monochromatic light) has a high overlap with the corresponding mode amplitude profile. Fortunately, the fundamental mode of a single-mode fiber has in most cases a profile which is close to that of a Gaussian beam (for robust guiding, with large enough *V* value), and Gaussian beams are well approximated by the outputs of most single-mode lasers. So the remaining task is

- to properly focus a laser beam such that the beam radius is close to that of the fiber mode,
- to place the fiber end at the beam focus (beam waist), and
- to align the fiber such that the beam focus hits the fiber core with proper orientation.

Obviously, the position error of the incident beam should be small compared with the mode radius. The following formula tells how the launch efficiency (disregarding possible reflections of the interface) depends on the position error Δ*x* and also on possible deviations between the input beam radius *w*_{1} and the mode radius *w*_{2}, if we can assume Gaussian mode profiles:

We see that for a perfect beam size, an offset of one beam radius reduces the coupling efficiency already to 1 / *e* ≈ 37%, and a 5 times smaller error allows for a 90% coupling efficiency. Note that the equation holds only for Gaussian profiles, but in most cases this is a good approximation.

The beam direction also has to be correct. This is not so sensitive, however, for typical single-mode fibers. The angle error should be well below the beam divergence, but that is relatively large for small more areas.

**The Effect of Imperfect Launch Conditions**

What happens, for example, if we somewhat misalign the input beam?

Figure 5 shows a simulated example, where the input laser beam is displaced by 1/10 of the beam radius. After some propagation length, only light in the guided mode remains. All other light is lost in the cladding. (The cladding/coating interface is often quite lossy.) At the end of a 10 cm long fiber, for example, one will find only light in the core, having a profile which is only determined by the mode profile. The launch conditions influence only the launched power, but not the output beam profile.

Figure 5: Light propagation at 1.5 μm wavelength in a single-mode fiber with displayed input beam. The numerical simulation has been done with the software **RP Fiber Power**.

## Multimode Fibers

Multimode fibers are fibers having multiple guided modes at the operation wavelength – sometimes only a few (→ few-mode fibers), but often many. The fiber core is often quite large – not much smaller than the whole fiber (see Figure 6). At the same time, the numerical aperture is often relatively high – for example, 0.3.

Figure 6: A single-mode fiber (left) has a core which is very small compared with the cladding, whereas a multimode fiber (right) can have a large core.

This combination leads to a large *V* number, which in turn leads to a large number of modes. For step-index fibers with large *V*, it can be estimated with the following formula, when counting both polarization directions:

Fibers with a smaller number of guided modes, e.g. with V numbers between 3 and 10, are sometimes called *few-mode fibers*.

Multimode fibers are required, if light with poor spatial coherence needs to be transported. For example, this is the case for the output of typical high-power laser diodes, such as diode bars. Whereas only a tiny portion of their output power could be launched into a single-mode fiber, very efficient launching is possible for a multimode fiber with sufficiently large core and/or high NA. Another example is the use of light-emitting diodes (LEDs) instead of laser diodes as cheap signal sources in fiber-optic links. Other applications exist imaging, for example; the transmission of image information requires devices with many spatial modes.

**Specifications for Multimode Fibers**

A basic specification of a multimode fiber contains the core diameter and the outer diameter of a multimode fiber. Common telecom fibers (fibers for optical fiber communications over moderate distances) are 50/125 μm and 62.5/125 μm fibers, having a core diameter of 50 μm or 62.5 μm, respectively, and a cladding diameter of 125 μm. Such fibers support hundreds of guided modes. Large-core fibers with even substantially larger core diameters of hundreds of micrometers are also available.

**Launching Light into a Multimode Fiber**

Compared with a single-mode fiber, a multimode fiber allows for much easier launching of light, particularly if it supports many guided modes. For efficient launching, one has to fulfill two conditions:

- The input light should essentially only hit the core, not the cladding.
- The input light should not contain significant amounts of power propagating with angles larger than arcsin NA.

If the *M*^{2} factor of the input light is sufficiently small, it is possible to fulfill these two conditions simultaneously. The maximum *M*^{2} factor for efficient launching of a beam with super-Gaussian profile can be estimated from the following formula:

This is actually the approximate beam quality factor from the fiber if the optical power is well spread over all modes. (The estimate is accurate only if the fiber has many guided modes.) Of course, efficient launching does not only require a low enough *M*^{2} factor, but also a suitable shape of the intensity profile in real space and Fourier space.

As an example, consider a fiber with 25 μm core radius and a numerical aperture of 0.2. Figure 8 shows the intensity profile of a monochromatic input beam at 1000 nm, which has been numerically constructed such that it just fills the fiber core and also has an angular distribution which reaches out up to the limit set by the numerical aperture of the fiber. The beam profile was essentially made by starting with a super-Gaussian intensity profile with entirely random phase values (leading to a huge divergence), then filtering with another super-Gaussian function in the Fourier domain, and once again applying the super-Gaussian filter in the spatial domain.

The angular distribution leads to the complicated intensity variations. Much smoother intensity profiles for the same beam quality are possible for non-monochromatic beams: although each wavelength component has a complicated profile, the fluctuations of these can average out to a smooth overall profile. (Particularly for non-monochromatic beams, a smooth intensity profile does *not* indicate a high beam quality.)

Figure 8: Intensity profile of a multimode beam which has about the maximum possible *M*^{2} value for efficient launching into the fiber.

The constructed beam turns out to have an *M*^{2} value of 12, which is not far below the limit of 15.7 as calculated from the formula mentioned above. Figure 9 shows how it propagates in the fiber. The beam profile undergoes strong changes in the fiber, but nearly all of the light stays guided.

Figure 9: Evolution of the intensity profile in the fiber. Only little light is lost to the cladding in the first few millimeters.

Similar simulations exhibit substantial launch losses if the initial beam size or the angular range is further expanded. Also, efficient launching requires a somewhat lower *M*^{2} value (below 10) if the beam profile is roughly Gaussian rather than super-Gaussian.

If one would launch light into specific high-order modes, that light could have a roughly 2 times larger *M*^{2} value than according to the formula above.

**Single-mode Propagation in Multimode Fibers**

If one launches light entirely into the fundamental mode of a multimode fiber, the beam profile should in principle stay unchanged during propagation. One would then obtain an output with high beam quality, similar to that of a single-mode fiber. However, various kinds of disturbances may lead to mode coupling: some light may be coupled into higher-order modes, so that the beam quality can be spoiled.

Fortunately, such coupling effects are often not that strong. As an example, consider a step-index fiber with 20 μm core diameter and an NA of 0.1. This is a few-mode fiber, supporting 6 guided modes (when counting all mode orientations). We take a 10 mm long piece of that fiber and introduce a relatively sharp bend, where the inverse bend radius rises smoothly to 1 / (10 mm) in the middle and back to zero again. That bending causes a substantial shift and deformation of the mode profile in the middle of the fiber:

Figure 10: The beam profile in the middle of a bent fiber is substantially shifted away from the center of the core.

Nevertheless, at the fiber end, where the bending has ended, the original beam profile is nearly unchanged; virtually all power remains in the LP_{01} mode:

Figure 11: Evolution of beam profile in a fiber which is bent in the middle part only. (The spatial coordinates do not reflect the bending, which is simulated as a linear addition to the index profile; one sees only the shift of the mode profile as a consequence of the bending.) The beam profile nicely comes back to the original position at the fiber end.

For fibers with large mode areas, the β values of different modes come much closer. Therefore, the beat length of two modes is much longer, and even relatively slowly changing disturbances can effectively couple modes. It is therefore more difficult in large mode area few-mode fibers to preserve single-mode propagation.

Source: rp-photonics

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