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Dispersion is a phenomenon in which the phase velocity and group velocity of light propagating in an optical fiber depend on the optical frequency. It is related to many applications of optical fiber. For example, it has a great influence on the propagation of telecommunication signals and the propagation of ultrashort pulses in optical fiber communication.

Origin of Dispersion

In fiber optics, dispersion comes from two completely different sources:

- Glass material has some material dispersion, eg applied to a plane wave propagating in a homogeneous sheet of that glass. This means that the refractive index is wavelength dependent.
- There is also waveguide dispersion: Since in fiber we don’t have plane waves (even though fiber modes usually have plane wavefronts), but rather spatially confined light waves, the dispersion is modified.

Understanding Waveguide Dispersion

Waveguide dispersion is easier to understand by considering a somewhat contrived situation: the fiber core is square rather than circular when viewed from the end face, with width a in the x and y directions. We also assume high exponential contrast, so at least the low-order modes have essentially no intensity outside the core. In this case, each mode field (inside the core) is essentially a superposition of four plane waves. Two have wave vector components in the ± x direction and two have wave vector components in the ± y direction.

Since the mode field must vanish at the edge of the core, the x-component of the wave vector must satisfy the condition k x a = j x π and a positive integer mode index j x. (kx is the phase advance per unit length in the x direction.) Similar rules apply for the y direction. Figure 1 shows the lateral amplitude distribution for j x = 3 and j y = 4.

Figure 1: “Amplitude distribution of modes in a square-core fiber.

For dispersion, what matters is the phase advance in the z direction. The phase constant is:

This follows from a simple calculation: we have a wavevector of certain magnitude, determined by the wavelength and refractive index n of the core, and this wavevector has transverse and longitudinal components that add up according to the Pythagorean law.

We now see that higher-order modes have lower phase constants β, mainly because their wave vector components are more strongly tilted with respect to the fiber axis. This means that their phase velocity vph = ω / β is higher than that of the fundamental (lowest order) mode.

The group velocity is the inverse of the derivative d β / d ω. Since each mode has a wavelength-independent transverse component, its β rises faster with frequency for higher-order modes. (As frequency increases, an increase in 2πn/ λ without increasing the transverse component means that the longitudinal component rises faster.) Thus, higher-order modes have lower group velocities.

Dispersion of Multimode Fiber

In real fibers, we typically encounter more complex situations due to circular symmetry, a possibly smooth index profile, and modes that extend significantly into the cladding. However, various basic aspects are basically the same as in the above examples. As a more realistic example, let’s look at the mode of a multimode germanium silica fiber. Figure 2 shows the exponential distribution and the radial mode function.

Figure 2: Modal function, index profile and radial profile of the effective index. Different colors are used for different l values.

Figure 2: Modal function, index profile and radial profile of the effective index. Different colors are used for different l values.

Figure 3 shows the group index for the schema. (The group velocity is c divided by the group index.) They rise at short wavelengths due to material dispersion. The two dashed gray curves show the core (at its center) and cladding values; the group index a of the mode is much higher than these two, already showing the additional influence of the waveguide dispersion.

Figure 3: Group index versus wavelength for all modes.

Figure 3: Group index versus wavelength for all modes.

Around the mode cutoff, the group index often drops, ie the group velocity rises. This is related to weaker pattern constraints in the regime. For modes with l = 0 (black curve), the group velocity can approach that of the envelope.

We can finally look at group velocity dispersion. This is essentially the frequency derivative of the group velocity. Figure 4 shows how the group velocity varies with wavelength for all modes. These values generally lie between the core and cladding values well below the mode cutoff wavelength and rise closer to the cutoff wavelength.

Figure 4: Group Velocity Dispersion vs. Wavelength for All Modes.

Figure 4: Group Velocity Dispersion vs. Wavelength for All Modes.

(See also a more complete description of the simulation model used.)

Of course, the dispersion properties depend on the detailed refractive index profile. This gives us the opportunity, for example, to tailor the dispersion properties of telecom fibers. This applies especially to single-mode fibers used in long-distance data transmission. For example, dispersion-shifted fibers with W-shaped refractive index profiles, where the zero-dispersion wavelength has been shifted from the 1.3-μm region (which is naturally the case for single-mode silica fibers) to the 1.5-μm region, modern telecommunication systems using erbium-doped fiber amplifiers where it runs. Zero dispersion is not always beneficial in practice. One also uses dispersion flattened fibers, with reduced wavelength dependence of group velocity dispersion, ie low higher order dispersion.

Unfortunately, an arbitrary dispersion distribution cannot be obtained. With common all-glass fibers, there are serious limitations. For example, anomalous dispersion (negative GVD) cannot be obtained in the visible wavelength range. Note also that one is concerned with various aspects such as effective mode area, bending losses, and manufacturing tolerances, and that trade-offs can be made.

Photonic crystal fibers containing tiny air holes can achieve a higher degree of dispersion control. The placement of the air holes provides a lot of room for optimization. Here, for example, anomalous dispersion in the visible wavelength range can be obtained. Note that calculations for such fibers are more difficult, have high index contrast, and are partly based on different light guiding principles.

Note, however, that none of these approaches have the potential to control dispersion in large mode area fibers. Such large modes always have phase constants (β values) close to the corresponding wavenumbers in the corresponding core material, and the influence of waveguide dispersion is very small. This is essentially because large modes have dispersion behavior similar to plane waves, with weaker diffraction and waveguide effects.

Chromatic Dispersion in Fiber Optic Links

One might think that dispersion is always bad for transmitting telecommunication signals in fiber optic links because it tends to spread and distort the signal in time. In fact, dispersion (as well as intermodal dispersion) introduces a dispersion power penalty, i.e. more optical power is required to achieve the same bit rate. Thus, for some time it seemed that one should operate fiber links in the 1.3-μm wavelength range close to the zero-dispersion wavelength (ZDW) of standard silica fibers, or use dispersion-shifted fibers with ZDW. 1.5-μm wavelength region where Erbium-doped fiber amplifiers can be used. However, it turns out that, especially when using WDM, it is desirable to have a certain amount of dispersion, as this mitigates nonlinear effects.

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