## An Apology

This blog post is rather long (3 parts). I have had so many questions on this topic lately that I thought I should put some of the notes into a more formal format. The discussion is very specific to fiber optic networks and requires some knowledge of fiber to follow.

## Introduction

I often work with people who are new to fiber optics and they often find dispersion confusing. Dispersion is caused by the variation of the speed of light in glass with wavelength, and the distortion it causes can limit the range of some deployments more than attenuation. People are very used to the idea that the speed of light is constant in a vacuum, but they are unaccustomed to the idea that the speed of light varies on a fiber as a function of wavelength, polarization, and fiber construction. In our system, dispersion is caused mainly by the variation in light speed as a function of wavelength, which is called chromatic dispersion.

Chromatic dispersion causes the pulses of light (“bits”) on the fiber to stretch out in time and reduce in amplitude. The stretching out is called Inter-Symbol Interference (ISI). The reduction in the amplitude of the pulses is called the dispersion Power Penalty (). Figure 1 illustrates how the pulse distorts as it moves down the fiber.

While doing some system modeling, I noticed that there were different equations being used to compute . Here are some examples of these equations.

(Eq. 1) | (see Ref 1, 4) | |

(Eq. 2) | (see Ref 5) | |

(Eq. 3) | (see Ref 2, 3) | |

(Eq. 4) | (Ref lost) |

where

*B*is the bit rate*L*is fiber distance*D*is the dispersion constant of the fiber- is the spectral standard deviation of the laser

I started to look at the equations in detail because I wanted to know why different equations were used to model the same thing. It turns out the equations are very similar and simply reflect different ways of defining or measuring some of the parameters critical to dispersion. The derivation of Equation 4 also uses a mathematical approximation. I thought it would be useful to document where the equations come from and why they look different. This example also illustrates a nice application of the normal curve that does not involve statistics.

## Background

#### Speed of Light and Index of Refraction

The key to understanding chromatic dispersion is to understand that the index of refraction in glass varies with wavelength. The relationship between the speed of light in glass and the index of refraction is given by the following equation.

where

*c*is the speed of light in a vacuum- is the speed of light in glass
- is the index of refraction
- is the wavelength of light.

Figure 2 illustrates how the index of refraction varies by wavelength and type of glass.

### Sources of Chromatic Dispersion

#### Fiber-Related

When we talk about chromatic dispersion, we are talking about a characteristic that is composed of three parts.

- Material Dispersion

This contribution is caused by the variation of the index of refraction in glass with wavelength. A prism uses this form of dispersion to separate out the colors of light. Material dispersion has nothing to do with the fiber – it is a property of the glass. Figure 1 illustrates how the index of refraction in various forms of glass varies with wavelength. - Waveguide Dispersion

The fiber is a form of waveguide and the optical power divides between the core and cladding. The cladding and core indexes of refraction are different, which causes dispersion. - Profile Dispersion

The glass within the core and cladding each have indexes of refraction that varies with wavelength and their construction. This also introduces dispersion.

Mathematically, chromatic dispersion is usually modeled by a single parameter that consists of three terms.

where

*D*is the total chromatic dispersion constant*D*is the material dispersion constant_{M}*D*is the waveguide dispersion constant_{W}*D*is the profile dispersion constant_{P}

The following discussion assumes that all the sources of chromatic dispersion can be modeled using the single parameter *D*.

#### Optical Source-Related

If the fiber was driven by a single wavelength, no chromatic dispersion would occur. However, no source of light produces a single wavelength – they all generate a range of wavelengths. In fact, simply generating a pulse causes some spectral spreading. Sometimes the dispersion is significant – sometimes it is not. The purest sources of light comes from lasers, and our systems are driven by lasers. There are two main types of lasers used in telecommunications: Fabry-Perot (FP) and Distributed Feedback (DFB). The FP lasers generate multiple discrete wavelengths (called modes) and are subject to a rather nasty form of dispersion called mode-partition noise. This imposes a severe limitation on the range of FP-based systems. I will not be discussing mode-partition noise here. Our systems use DFB lasers, which generate light in very limited band with a distribution that is modeled well by a normal curve. The graphs in Table 1 illustrate the spectral characteristics of these lasers.

FP Laser |
DFB Laser |

Source | Source |

Part 2 of this blog will address the analysis and modeling of dispersion losses.

## References

1. Agrawal, Govind. P.J. Anthony, and T.M Shen. “Disperson Penalty for 1.3-µm Lightwave Systems with Multimode Semiconductor Lasers.” Journal of Lightwave Technology. May 1988: pp 620-625. Print.

2. Agrawal, Govind. Fiber-Optic Communication Systems. 3rd ed. NY: Wiley, 2002. p 204. Print.

3. Keiser, Gerd . Optical communications Essentials. 1st. Boston, MA: McGraw-Hill Professional, 2003. p. 265. Print.

4. Miller, John, and Ed Friedman. Optical Communications Rules of Thumb. Boston, MA: McGraw-Hill Professional, 2003. p. 325. Print.

5. Agrawal, Govind. Lightwave technology: Telecommunication systems, Volume 2. New York: Wiley, 2005. p. 170. Print.

Pingback: Optical Fiber Dispersion Formula – Where Did This Come From? | Math Encounters Blog

Pingback: A Little Specification Reading Before Bed | Math Encounters Blog

This may be a stupid qustion, but I did not find the answer even after a quite long search on the web, so perhaps you can help me out:

As far as I understand, if I send a single pulse signal with a laser diode or LED, there should be two possible sources of a spread in the wave length. One is the spread that is necessary simply due to the fact that a Fourier transform of the signal has a frequency distribution, the other is due to the laser/LED not being exactly monochromatic. IIUC, In this (and the following entry) you only write about the width of the light source. Are both effects included in this?

And what is the relative size of these two effects in practice? If I send a signal at 1GB/s, I would expect the Fourier transform to have a width of 1GHz; if I have a light signal at a few hundred nanometers with a spectral width of a few nanometers, the frequency difference is about 1e12Hz. Is this correct, implying that the uncertainty in the laser frequency is usually much larger than the Fourier width?

The equations I use are based on (1) the laser having a small spectral width (e.g. 0.1 nm), and (2) the signal has its information content contained in a frequency range that is related to the data rate (e.g. 90% of the signal power is contained within 1.5 x bit rate). For today’s technology, the frequency content of the data is contained in bandwidths measured in GHz. I never worry about the actual frequency of the light, which is really a carrier of very high frequency (e.g. THz, as you state). We do routinely talk about the difference between two closely spaced WDM wavelengths in terms of GHz (e.g. see this table of ITU wavelengths on a 100 GHz grid).

In your example, 1 Gbps signal has a bandwidth of ~1 GHz. We do see the laser wavelength shift when it is being driven with data (~0.2 nm). There are numerous causes for the wavelength shift. One of the largest shift components is a phenomenon called chirp. Chirp is caused when a laser is being driven and charge carriers are injected into or removed from the optical channel (this cannot be prevented). This changes the effective index of refraction and alters the wavelength of the laser. Chirp is a far larger effect than anything having to do with the frequency of the light itself.

Like electrical signals, I am sure the true optical spectra is somehow related to a convolution of the carrier spectra and the signal spectra. However, it is a small effect relative to things like the the information spectra and chirp shift.

Mathscinotes

Thanks a lot, that was very helpful (and you gave me the right keywords for some further reading…). I have one follow-up question, though: In part 2, you use sigma_lambda, but I’m a bit confused by that. First you say that this is the sdev of the pulse width, later you write that it is the spectral width of the laser. Shouldn’t it be rather the spectral width of the signal (including the spectral width of the signal I would have even if my laser were monochromatic and the width due to the laser being non-monochromatic and having chiprping)? Because otehrwise, from the formula there should be no dispersion if the laser is monochromatic?

The formulas assume that the laser’s spectrum can be modeled using a Gaussian shape. This shape is characterized by a standard deviation, which I called σ

_{λ}. There is a relationship between spectral width and σ_{λ}, which I document here. Ideally, σ_{λ}would include all sources of wavelength variation. I sometimes use spectral width and σ_{λ}interchangeably, however, they are different. They are related by a constant factor. I will update the blog post when I have time.Mathscinotes

Pingback: Mathcad Example Using Decibels | Math Encounters Blog