Dispersion Power Penalty Modeling (Part 1)

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.

Figure 1: Illustration of Pulse Distortion 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.

Figure 2: Refractive Index as a Function of Wavelength and Glass Type

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_M is the material dispersion constant
• D_W is the waveguide dispersion constant
• D_P is the profile dispersion constant

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.

Table 1: Examples of Laser Spectral Characteristics

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.