# Origins of Chromatic Dispersion

This is a continuation from the previous tutorial - telephoto lenses.

### Introduction

An optical fiber’s dispersion is the tendency for the fiber to either broaden or  arrow a pulse as it travels along the fiber. The term chromatic dispersion is used to refer to the change in pulse shape that results when the velocity of the signal power along the fiber length is a function of optical frequency or wavelength.

Chromatic dispersion alters the pulse shape because the signal power has a finite spectral width due to the spectral width of the signal modulation and the spectral width of the laser.

The different signal frequencies or wavelengths of the signal travel along the fiber at different velocities, causing a digital pulse to spread in time, or an analog signal to become distorted.

In a single-mode fiber, chromatic dispersion of the fundamental mode is caused by the dispersive properties of the materials that the fiber is made from, referred to as material dispersion, as well as by the dispersive properties of the waveguide, referred to as waveguide dispersion.

### Material Dispersion

As light travels through a material, it is slowed relative to its speed in vacuum, $$c$$, by the factor $$1/n$$, where $$n$$ is the refractive index, because the electromagnetic wave interacts with the bound electrons in the material.

Because the bound electron oscillations have resonance frequencies that are characteristic of the material, the interaction between the electromagnetic wave and the electrons is frequency dependent. This results in a frequency dependent index of refraction which gives rise to chromatic dispersion.

The frequency-dependent relationship for the index of refraction of a material can be obtained from a simple classic model that treats the bound electrons as harmonic oscillators. The well-known Sellmeier equation for the frequency dependence of the index of refraction can be derived from an oscillator model.

The Sellmeier equation expressed in terms of optical wavelength is

$\tag{2.21}n^2-1=\sum_{j=1}^M\frac{\lambda^2B_j}{\lambda^2-\lambda_j^2}$

where $$M$$ is the number of electron resonances, $$\lambda_j$$ are the wavelengths of the electron resonances, and $$B_j$$ are constants obtained experimentally by the fitting to dispersion measurements.

Values for the coefficients of the Sellmeier equation for fused silica for a three-term fit to experimental data are shown in Table 2.1. Figure 2.8a shows the index of refraction as a function of wavelength for fused silica, germanium-doped silica, and fluorine-doped silica obtained using the Sellmeier coefficients given in Table 2.1.

The group delay per unit length, $$\tau$$, of a wave propagating along a fiber is given by

$\tag{2.22}\tau=\frac{1}{c}\frac{\text{d}\beta}{\text{d}k_0}$

where $$\beta=n_\text{eff}k_0$$, $$n_\text{eff}$$ is the mode effective index, and $$k_0$$ is the free space propagation constant.

By substituting $$\beta=2\pi{n}(\lambda)/\lambda$$ into Eq. (2.22), we can recast Eq. (2.22) to express the group delay in terms of the index of refraction and its wavelength derivative

$\tag{2.23}\tau=\frac{1}{c}\left(n-\lambda\frac{\text{d}n}{\text{d}\lambda}\right)$

We can see clearly from Eq. (2.23) that the group delay is wavelength dependent when the index of refraction varies with wavelength.

Figure 2.8b plots as a function of wavelength the length normalized group delay for silica, germanium-doped silica, and fluorine-doped silica calculated from Eq. (2.23) using the index of refraction values plotted in Fig. 2.8a.

Pulse distortion results from the dependence of group delay with wavelength because the spectral components of a pulse will experience varying delays as they propagate along a fiber.

The material dispersion of the fiber is given by wavelength derivative of the group delay

$\tag{2.24}\frac{\text{d}\tau}{\text{d}\lambda}=-\frac{\lambda}{c}\frac{\text{d}^2n}{\text{d}\lambda^2}$

Figure 2.8c shows the material dispersion curves for silica, germanium-doped silica, and fluorine-doped silica calculated using Eq. (2.24) and the index of refraction curves in Fig. 2.8a.

It is important to note that the material dispersion of silica is zero at about 1270 nm, and it is, therefore, possible to make silica-based fibers with low dispersion in this wavelength region.

### Waveguide Dispersion

The chromatic dispersion of a step-index single-mode fiber resulting from waveguide effects is referred to as waveguide dispersion. Solutions to the Helmholtz equation for the electric fields in the cylindrical waveguide and the associated eigenvalues, $$u$$, $$w$$, and $$\beta$$, are wavelength dependent.

Recall that $$\beta$$ and $$\beta_t$$ must obey the relationship $$\beta_{t1}=(n_1^2k_0^2-\beta^2)^{1/2}$$. Therefore, even if $$n_1$$ is assumed to be constant (i.e., there is no material dispersion), the wavelength dependence of $$k_0=2\pi/\lambda$$ results in variation of $$\beta$$ and $$\beta_t$$ with wavelength.

By recasting Eq. (2.22) for group delay in terms of the wavelength derivative of $$\beta$$,

$\tag{2.25}\tau=-\frac{\lambda^2}{2\pi{c}}\frac{\text{d}\beta}{\text{d}\lambda}$

we see that the wavelength dependence of $$\beta$$ results in variation in group delay with wavelength.

Waveguide dispersion in a single-mode fiber can be understood through the following heuristic discussion of how the fundamental mode longitudinal propagation constant, $$\beta$$, changes with wavelength.

We consider two bounding cases: first, where the optical wavelength is large relative to the core diameter, and second, where it is small relative to the core diameter.

For the case of very long wavelength, the fundamental mode is very loosely confined to the core and the ratio of the power carried in the cladding to the total power approaches unity.

At very long wavelength, the longitudinal propagation constant of the fundamental mode, $$\beta$$, approaches that of a plane wave propagating in the cladding, $$kn_\text{clad}$$.

The group delay will approach that of a plane wave propagating in the cladding, $$\beta$$ approaches its lower limit for a bound mode, $$kn_\text{clad}$$, and the group velocity is maximized. The group delay asymptotically approaches the silica curve in Fig. 2.8b at long wavelength.

Now considering the case of very short wavelength, the fundamental mode is very tightly confined to the core and the ratio of power carried in the cladding to the total power approaches zero.

In this case, the longitudinal propagation constant, $$\beta$$, approaches that of a plane wave propagating in the core material, $$kn_\text{core}$$. In this case as $$\beta$$ approaches its upper limit, $$kn_\text{core}$$ and the group velocity is minimized.

Because for most step-index fibers the core material is germanium-doped silica, for the very short wavelength case the group delay approaches that of the curve for germanium-doped silica plotted in Fig. 2.8b.

The finite element method was used to solve the scalar Helmholtz equation and determine the dispersion properties of the fundamental mode for a step-index, single-mode fibers.

Figure 2.9 shows the material dispersion, waveguide dispersion, and total dispersion for a typical first-generation matched cladding single-mode fiber, with 4.65 microns core radius and core $$\Delta=0.34\%$$.

The material dispersion zero is at 1.28 microns. The core dimensions of the fiber have been chosen to provide the necessary waveguide dispersion so that the zero crossing of the total dispersion (material + waveguide) is located at the 1.31-micron local attenuation minimum.

With this design choice, the dispersion zero and local attenuation minimum are collocated at 1.3 μm. Figure 2.10 shows the dispersion curves for step-index fibers with core radii ranging from 2.0 to 5.0 microns.

As the core radius decreases, the magnitude of the waveguide dispersion curve is seen to increase in absolute value, while the change in the material dispersion is small.

The location of the zero of the total dispersion curve, therefore, is shifted from around 1.3 μm at the larger core radii to around 1.55 μm at the smaller radii. This set of design choices results in the total dispersion zero to be collocated with the absolute loss minimum at 1.55 μm.

It was recognized that the variation of the waveguide dispersion with wavelength could be tailored to provide total dispersion curves with flattened shapes, multiple zero crossings, or reduced dispersion slope in the 1.55 μm transmission window by proper design of multilayer index of refraction profiles.

These profiles typically have a central core region with the highest delta, a surrounding region with index reduced close to or below the cladding level, and a third concentric layer with raised index, typically at a level between the first two layers.

Computational techniques, such as FEM, are required to obtain quantitative values for the properties of these complicated waveguide structures. However, the previous heuristic discussion can be applied to gain insight into how the multilayer waveguides behave.

Consider a fiber with central core surrounded by a fluorine-doped depressed-index trench and then a raised index ring. At very short wavelength, the electric field is tightly confined within the central layer and the overall group delay and dispersion approaches that of a plane wave propagating within this region.

As wavelength increases, the mode starts to extend more into the trench region. Because of the depressed index of refraction of the trench, a plane wave traveling in this region has faster velocity and lower group delay relative to one traveling within the core.

As the mode extends into the trench, the group delay and dispersion properties start to tend toward those of the trench region. The larger the contrast in the group velocity between the central region and the trench region, the greater is the ability to tailor the group delay and dispersion of the waveguide.

The presence of the third raised index layer adds an additional guiding layer to the waveguide and as wavelength increases further, when overall structure is properly designed the energy will spread across the entire three layer structure and remain confined.

As the mode spreads out over the entire structure, the fraction of energy contained within the trench region peaks and eventually decreases as the ring layer provides guidance.

With appropriate choices of waveguide dimensions, the waveguide can be designed so that as the mode grows with wavelength and extends outward into the trench and ring regions, the group delay and the magnitude and shape of the waveguide dispersion curve of the mode can be tailored to provide total dispersion multiple zeros, or flattened shape.

Multilayer index profiles with extreme contrast in index between the core and trench can provide very high values of waveguide dispersion (e.g., -150 ps/nm-km, for use as dispersion compensation devices).

The next tutorial discusses about polarization mode dispersion.