# Dispersion in Fibers

This is a continuation from the previous tutorial - Attenuation in Fibers.

Dispersion is the primary cause of limitation on the bandwidth of the transmission of optical signals through an optical fiber. There are waveguide and modal dispersions in an optical waveguide in addition to material dispersion.  Both material dispersion and waveguide dispersion are examples of chromatic dispersion because both are frequency dependent. Waveguide dispersion is caused by frequency dependence of the propagation constant β of a specific mode due to the waveguiding effect. The combined effect of material and waveguide dispersions for a particular mode alone is called intramode dispersion. Modal dispersion is caused by the variation in propagation constant between different modes: it is also called intermode dispersion. Modal dispersion appears only when more than one mode is excited in a multimode fiber. However, it exists even when chromatic dispersion disappears. In contrast, if only one mode is excited in a fiber, only intramode chromatic dispersion has to be considered even when the fiber is a multimode fiber.

Material Dispersion

The physical mechanism responsible for material dispersion is discussed in the material dispersion tutorial. For optical fibers, the materials of interest are pure silica and doped silica. We first consider the characteristics of relevant parameters for these materials using the general mathematical definitions given in the phase velocity, group velocity, and dispersion tutorial. The parameters of interest are the index of refraction, n, the group index, N, and the group-velocity dispersion, D. Although it is more natural to consider the propagation constant, k, or β in a waveguide, and its derivatives as a function of frequency, ω, in practice these parameters are commonly given as a function of the free-space wavelength, λ.

The index of refraction of pure silica in the wavelength range between 200 nm and 4 μm is given by the following empirically fitted Sellmeier equation:

$\tag{96}n^2=1+\frac{0.6961663\lambda^2}{\lambda^2-(0.0684043)^2}+\frac{0.4079426\lambda^2}{\lambda^2-(0.1162414)^2}+\frac{0.8974794\lambda^2}{\lambda^2-(9.896161)^2}$

where λ is in micrometers.

As discussed, the index of refraction can be changed by adding dopants to silica, thus facilitating the mans to control the index profile of a fiber. The amount of index change depends on the type and concentration of dopant or dopants. Specifically, doping with germania or alumina increases the index of refraction. Therefore, the coefficients in (96) actually depend on the composition of the glass.

The indices of refraction as functions of wavelength for pure silica and a germania-silica glass that has 13.5 mol % GeO2 and 86.5 mol % SiO2 are shown in figure 9(a) below. The group index N and the group-velocity dispersion D can be calculated using (171) and (172), respectively [refer to the phase velocity, group velocity, and dispersion tutorial]. The group indices for both glasses are also shown in figure 9(a), while the group-velocity dispersion is shown in figure 9(b).

It can be seen that the addition of GeO2 to silica not only increases the index of refraction, but also increases material dispersion. As a result, the point of zero material group-velocity dispersion is shifted from 1.284 μm for pure silica to 1.383 μm for germania-silica glass. This increase in index of refraction and in dispersion is reduced if the percentage of GeO2 is reduced.

The effects of other dopants vary. For example, doping with 9.1 mol % P2O5 increases the index of refraction by more than 1% but only slightly shifts the dispersion curve, whereas doping 13.3 mol % B2O3 results in a reduction of the index of refraction by less than 0.4% but shifts the point of zero dispersion to 1.231 μm. Clearly, it is possible to control the modification of material dispersion by dopants. Figure 9  (a) Index of refraction n and group index N and (b) group-velocity dispersion D as functions of wavelength for pure silica (solid curves) and germania-silica containing 13.5 mol % GeO2 (dashed curves). Zero group-velocity dispersion appears at 1.284 μm for pure silica.

Waveguide Dispersion

The propagation constant of a guided mode of a fiber is determined both by the parameters of the fiber, such as its index profile and core size, and by the material properties. Therefore, the frequency dependence of β of a particular mode has mixed contributions from material dispersion and waveguide dispersion.

It is in fact more convenient to consider this combined effect directly. To do so, we only have to replace k in all of the formulas in the phase velocity, group velocity and dispersion tutorial by β of the particular mode under consideration, thus defining the effective refractive index nβ, the effective group index Nβ, and the effective group-velocity dispersion Dβ for the mode:

$\tag{97}n_\beta=\frac{c\beta}{\omega}$$\tag{98}N_\beta=c\frac{\text{d}\beta}{\text{d}\omega}=n_\beta-\lambda\frac{\text{d}n_\beta}{\text{d}\lambda}$

and

$\tag{99}D_\beta=c\omega\frac{\text{d}^2\beta}{\text{d}\omega^2}=\lambda^2\frac{\text{d}^2n_\beta}{\text{d}\lambda^2}$

The exact frequency dependence of these parameters depends on the parameters of the fiber, which are, specifically, the V number, the normalized index difference Δ, and, in the case of the power-law profiles, the parameter $$\alpha$$. Since most optical fibers are weakly guiding, we consider only weakly guiding fibers in the following to simplify the mathematics.

In the case of a step-index fiber, it is convenient to use the normalized guide index b, which has the same form as that defined in (47) [refer to the step-index planar waveguides tutorial], for the step-index planar waveguide:

$\tag{100}b=\frac{n_\beta^2-n_2^2}{n_1^2-n_2^2}$

Taking the weakly guiding approximation of (48) [refer to the weakly guiding fibers tutorial] and using (97), we have

$\tag{101}n_\beta\approx n_2(1+b\Delta)$

The b parameter can be found by solving (57) [refer to the weakly guiding fibers tutorial] together with (11) and (12) [refer to the step-index fibers tutorial] for β. It is plotted as a function of the fiber V number in figure 10 below for some LP modes. Figure 10  Normalized propagation constant b as a function of fiber V number for some LP modes of a weakly guiding step-index fiber

The frequency or wavelength dependence of nβ can be found from the dependence of b on V. Using (98) and (99), we also find that

$\tag{102}N_\beta\approx N_2\left[1+\frac{\text{d}(Vb)}{\text{d}V}\Delta\right]$

where N2 is the group index of the cladding material of the fiber, and

$\tag{103}D_\beta\approx D_2\left[1+\frac{\text{d}(Vb)}{\text{d}V}\Delta\right]+\frac{N_2^2}{n_2}\frac{V\text{d}^2(Vb)}{\text{d}V^2}\Delta$

where D2 is the group-velocity dispersion of the fiber cladding.

In deriving (102) and (103), terms such as dΔ/dω and d2Δ/dω2 that contain the differential material dispersion are dropped because they are usually very small compared with the terms we keep. For more accurate calculations, they should be included.

In each of the relations in (101)-(103), the first term is the material contribution while the other terms are the waveguide contribution. The waveguide group delay parameter, d(Vb)/dV, and the waveguide dispersion parameter, Vd2(Vb)/dV2, are plotted as a function of fiber V number in figure 11(a) and 11(b), respectively. Figure 11  (a) Waveguide group delay parameter, d(Vb)/dV, and (b) waveguide dispersion parameter, Vd2(Vb)/dV2, as a function of fiber V number for some LP modes of a weakly guiding step-index fiber.

It can be seen from the discussion above and from the data plotted in figure 10 that nβ is bounded by n1 and n2, reaching n2 near cutoff and approaching n1 far away from cutoff. In contrast, figure 11(a) shows that only LP0n and LP1n modes have nβ reaching N2 at cutoff because only LP0n and LP1n modes have their power moved completely away from the core into the cladding at cutoff. An LPmn mode with m ≥ 2 still has a large fraction of its power concentrated in the core at cutoff as discussed in the weakly guiding fibers tutorial.

Figure 11(b) shows that the group-velocity dispersion can be modified by the waveguide contribution. As a practical example, figure 12 shows the combined material and waveguide contributions for the fundamental mode of a step-index germania-silica fiber with an index step of n1 - n2 = 0.006 and a core radius of a = 3 μm. The core is assumed to have 13.5 mol % GeO2 for n1, N1, and D1 to be consistent with those of the material properties of the germania-silica glass shown in figure 9. It is also assumed that the differential material dispersion, dΔ/dω, is negligible so that D1 = D2 to show the effect of waveguide dispersion clearly. As shown in figure 12(b), the point of zero dispersion is shifted from that of the germania-silica material at 1.383 μm to 1.5 μm because of the waveguide contribution. Figure 12  (a) Effective index of refraction and group index and (b) group-velocity dispersion of the fundamental mode as a function of wavelength. The solid curves show the effective parameters of the mode with both material and waveguide contributions. The dashed curves show only the material contribution to the core and cladding regions, labeled 1 and 2, respectively.

Example

A step-index single-mode fiber for transmitting a signal at λ = 1.35 μm is 100 km long. At this wavelength, the fiber has the following parameters for its silica cladding: n2 = 1.446, N2 = 1.466, and D2 = -0.0027. Its core has a radius of a = 4 μm and an index of n1 = 1.450. What are the propagation constant, the group velocity, and the group-velocity dispersion of the signal propagating as the guided mode of the fiber? If a 10-ps pulse that has a spectral width of Δλps = 2 nm is sent through the fiber, what is its transmission time through the fiber? What is the pulse duration when it arrives at the other end of the fiber?

With the given parameters, we find that V = 2 at λ = 1.35 μm. Thus, this fiber is indeed a single-mode fiber for this wavelength. From figure 10 and 11, we find the following parameters for V = 2:

$b=0.416,\qquad\frac{\text{d}(Vb)}{\text{d}V}=1.065,\qquad\frac{V\text{d}^2(Vb)}{\text{d}V^2}=0.461$

Using these parameters, together with the given values of n2N2, and D2, we find from (101), (102), and (103) that

$n_\beta=1.448,\qquad N_\beta=1.470,\qquad D_\beta=-0.00082$

From these results, we then find the following parameters for the mode:

\begin{align}&\beta=\frac{2\pi n_\beta}{\lambda}=6.739\,\mu\text{m}^{-1},\qquad v_g=\frac{c}{N_\beta}=2.04\times10^8\,\text{m s}^{-1},\\&D_{\lambda}=-\frac{D_\beta}{c\lambda}=2.02\,\text{ps km}^{-1}\text{nm}^{-1}\end{align}

The transmission time of the pulse through the 100 km length of the fiber is

$t_\text{tr}=\frac{l}{v_g}=490\,\mu\text{s}$

The spread of the pulse due to group-velocity dispersion is

$\Delta t_g=|D_\lambda|\Delta\lambda_\text{ps}l=404\;\text{ps}$

Therefore, the pulse arrives at the end of the fiber after 490 μs with a substantially broadened pulse duration of Δtps = 10 ps + 404 ps = 414 ps.

Although the data shown in figures 10-12 are specific for step-index fibers, the formulas obtained in (101)-(103) are equally applicable to graded-index fibers. However, in order to use (101)-(103) for graded-index fiber, exact solution of the eigenvalue equation (75) [refer to the graded-index fibers tutorial] has to be carried out to obtain the frequency dependence of the propagation constant β, and thus the dependence of b on V, for each guided mode of interest. This would be the required procedure if the fiber under consideration were a single-mode graded-index fiber or a multimode graded-index fiber that supported only a few modes.

For a multimode graded-index fiber that supports a very large number of modes, the approximate solution of β given by (88) [refer to the graded-index fibers tutorial] can be used. Then, instead of expressing the index and dispersion parameters in terms of b and V, we can use (97)-(99) directly to obtain

$\tag{104}n_\beta=n_1(1-2\zeta\Delta)^{1/2}$
$\tag{105}N_\beta\approx N_1\left(1+\frac{\alpha-2-\delta}{\alpha+2}\zeta\Delta+\frac{3\alpha-2-2\delta}{\alpha+2}\frac{\zeta^2\Delta^2}{2}\right)$

and

$\tag{106}D_\beta\approx D_1\left(1+\frac{\alpha-2-\delta}{\alpha+2}\zeta\Delta\right)-\frac{N_1^2}{n_1}\frac{2(\alpha-\delta/2)(\alpha-2-\delta)}{(\alpha+2)^2}\zeta\Delta$

where

$\tag{107}\zeta=\left(\frac{M_\beta}{M}\right)^{\alpha/(\alpha+2)}$

and

$\tag{108}\delta=\frac{2n_1}{N_1}\frac{\omega}{\Delta}\frac{\text{d}\Delta}{\text{d}\omega}=-\frac{2n_1}{N_1}\frac{\lambda}{\Delta}\frac{\text{d}\Delta}{\text{d}\lambda}$

Again, the first term in each of (104)-(106) represents the material contribution, while the other terms account for waveguide contributions.

Modal Dispersion

Modal dispersion exists because different modes in a multimode waveguide propagate at different group velocities, as indicated by (105). Note that Dβ given by (106) is the total intramode dispersion including material and waveguide contributions for a mode that has a propagation constant β in a multimode fiber. It has nothing to do with intermode dispersion. To find the modal dispersion, we have to consider the difference in Nβ between modes of different β. This difference exists even when there is no intramode chromatic dispersion so that Dβ vanishes.

In a multimode fiber, the modal dispersion between the fundamental mode and the highest-order mode supported by the fiber can be estimated. Because the fundamental mode HE11, or LP01, has two-fold degeneracy, it corresponds to Mβ = 2. For the highest mode, Mβ = M. Therefore, for a fiber with a very large number of modes, ζlow = 2/M ≈ 0, while ζhigh = 1. They determine the minimum and maximum values of Nβ among the modes. The modal dispersion can then be expressed as

$\tag{109}N_\text{high}-N_\text{low}=N_1\left(\frac{\alpha-2-\delta}{\alpha+2}\Delta+\frac{3\alpha-2-2\delta}{\alpha+2}\frac{\Delta^2}{2}\right)$

Note that Nhigh > Nlow when α > 2 + δ, but Nhigh < Nlow when α < 2 + δ. Because group velocity vg = c/N, this dispersion represents the difference in the group velocity between different modes. Although it is always true that a low-order mode has a larger β and thus a smaller phase velocity than a high-order mode, the relationship between their group velocities is less straightforward. It depends on many factors, including the waveguide structure, the index profile, the material  properties, and how far away the modes are from cutoff. For example, it can be seen from (109) that a low-order mode travels faster than a high-order mode if α > 2 + δ, whereas the reverse is true if α < 2 + δ. In addition, it has to be kept in mind that even this statement is not always true for modes near cutoff, as can be seen from the discussions for step-index fibers and from figure 11(a) using (102). Therefore, modal dispersion can also be modified by choosing appropriate waveguide and material parameters.

Dispersion Compensation

We have seen that all three types of dispersion in a fiber can be modified to a certain extent by various means. Therefore, it is possible to engineer a desired dispersion characteristic through careful choice of the type and concentration of dopants to control the material dispersion while designing the fiber parameters to adjust the waveguide dispersion and the modal dispersion.

In some special applications, one might want a certain nonzero value of positive or negative dispersion at a particular wavelength. For example, one would need a finite amount of positive group-velocity dispersion in the application of fiber-grating compression of optical pulses, whereas one would need finite negative group-velocity dispersion for the generation and propagation of soliton pulses in a fiber.

Nevertheless, in most applications using fibers to transmit optical signals, dispersion in a fiber causes undesirable spreading of the signal, limiting the bandwidth of transmission. It is desirable to reduce the dispersion to zero, if possible, for such applications.

For applications that require the largest bandwidths, single-mode fibers are the choice because modal dispersion does not exist in a single-mode fiber. Because the zero-dispersion point of pure silica is near the window of a local minimum of attenuation at 1.3 μm, transmission systems based on this wavelength have the combined advantage of low attenuation and low dispersion and are a choice for long-distance optical communications.

As discussed in the preceding tutorial, the real minimum of attenuation appears at 1.55 μm wavelength. Therefore, it is usually desirable to shift the point of zero dispersion to this wavelength. This task can be accomplished by a combination of choices of dopants and waveguide parameters, as demonstrated by the example shown in figure 12 where the point of zero dispersion is shifted to 1.5 μm already. Such fibers are known as dispersion-shifted fibers. Zero dispersion in both 1.3- and 1.55-μm windows can also be accomplished by special profiling of the fiber, resulting in so-called dispersion-flattened fibers that have low dispersion in the region between 1.3 and 1.55 μm with zero crossings at both wavelengths. Figure 13 below shows an example of a dispersion-flattened fiber. Figure 13  (a) Index profile and (b) dispersion characteristics of a dispersion-flattened fiber

For multimode fibers, modal dispersion dominates intramode waveguide dispersion. It is then important to minimize the modal dispersion. From (109), it can be seen that modal dispersion can be minimized if we choose

$\tag{110}\alpha=2+\delta$

The value of δ depends on the dopants and the optical wavelength. In the near infrared spectral region, it is usually within the range of ±0.3 for most dopants. Therefore, the optimum profile for a low-dispersion multimode fiber is one close to a quadratic graded-index profile. This results in a modal dispersion of

$\tag{111}N_\text{high}-N_\text{low}=N_1\frac{\Delta^2}{2}$

In comparison, a step-index multimode fiber has $$\alpha=\infty$$ and

$\tag{112}N_\text{high}-N_\text{low}=N_1\Delta$

Because Δ is a very small number, the modal dispersion in an optimized graded-index fiber is substantially lower than that in a step-index multimode fiber.

it is interesting to see that the total intramode dispersion Dβ given by (106) is also mode dependent. However, when α is chosen to be the optimum value given by (110),  Dβ = D1, and the intramode dispersion becomes mode independent, indicating that waveguide dispersion is minimized. Therefore, a graded-index fiber that has a minimum modal dispersion also has a minimum waveguide contribution to the intramode dispersion for each individual mode.

Example

A multimode fiber of 10 km length has a core group index of N1 =  1.5 and an index step of Δ = 2%. If an optical signal sent through this fiber is carried by all of its guided modes, what is the transmission time of the signal? What is the temporal broadening of the signal due to modal dispersion if the fiber is a dispersion-optimized graded-index fiber? What is the broadening if the fiber is a step-index fiber?

The transmission time of the signal is

$t_\text{tr}=\frac{l}{v_g}=\frac{l}{c}N_\beta\approx\frac{l}{c}N_1=50\,\mu s$

In the case of an optimized graded-index fiber, the temporal broadening due to modal dispersion is

$\Delta t_\text{mode}=\frac{l}{c}(N_\text{high}-N_\text{low})=\frac{lN_1}{2c}\Delta^2=10\,ns$

In the case of a step-index fiber, the temporal broadening due to modal dispersion is

$\Delta t_\text{mode}=\frac{l}{c}(N_\text{high}-N_\text{low})=\frac{lN_1}{c}\Delta=1\,\mu s$

Clearly, temporal broadening of the signal due to modal dispersion is much worse in a step-index fiber than in an optimized graded-index fiber.

The next part continues with the Coupled-Wave Theory tutorial.