Passive Fiber Optics

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.

Generally, one might expect that a smaller core implies a smaller fiber mode. For a constant V number, this is true; the index contrast is then getting smaller for a larger core. However, if we keep the numerical aperture constant, the V number changes with the core radius, and the mode radius shows a non-monotonic dependence on the core radius, as can be seen in Figure 2 for an NA of 0.1:

One can see that for core radii below ≈4.8 μm, where the V number gets rather small, the mode radius increases. In the regime of small V values, the mode extends far beyond the core, and it deviates substantially from a Gaussian shape.

Figure 3 shows the same for a higher NA of 0.3:

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 μm². That is not too far from the data of the often used SMF-28 telecom fiber from Corning.

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 3 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 LP₀₁ mode, it then also supports LP₁₁ modes (actually two of these with orthogonal orientation). Below 787 nm, the LP₀₂ mode comes in additionally.

In principle, the fiber stays single-mode for any wavelength above the LP₁₁ 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.

For some applications, it is desirable to have rather large mode areas while still having single-mode guidance. For example, one may want to minimize nonlinear effects or maximize the amount of stored energy in pulsed fiber amplifier while preserving a high beam quality.

In principle, it is easy to obtain single-mode guidance even for a rather large core: one only has to reduce the index contrast (thus the numerical aperture). For example, one could increase the core radius in the above example fivefold to 20 μm. If we reduce the numerical aperture by the same factor to 0.024, we still obtain single-mode guidance – now with an active area of 1869 μm², which is 5² = 25 times larger than before. Unfortunately, however, that brings with it some trouble:

• As the index contrast is now extremely weak (0.0002), the fiber becomes extremely sensitive to small random index changes resulting from not perfectly controlled fabrication conditions. The NA of 0.024 is considered impractically small for currently available technology.
• Even if a perfect fiber is made based on that design, it will be very sensitive to bending. Figure 2 shows a numerical simulation, where the inverse bend radius grows linearly along the fiber. On the right side, a bend radius of 1 m is reached. Before that happens, the light experiences serious bend loss. This means that the fiber could be used only if it were essentially kept straight, and if any significant microbends could be avoided.

More advanced fiber designs have been developed, where mode areas well above 1000 μm² can be obtained in combination with better bending properties and lower sensitivity to fabrication errors. However, the described problems cannot be fully avoided. The fundamental problem is that a very large mode is necessarily only weakly guided, and thus sensitive to various additional effects.

Efficient launching light into a large mode area single-mode fiber is simpler than for a small area, as far as the position alignment is concerned. Note, however, that as the beam divergence gets smaller, the angular alignment becomes more sensitive.

Go to Part 4: Multimode Fibers or back to the start page.

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