Dispersion

Dispersion is the spreading out of light pulses as they travel along a fiber. It occurs because the speed of light through a fiber depends on its wavelength and the propagation mode. The differences in speed are slight, but like attenuation, they accumulate with distance. The four main types of dispersion arise from multimode transmission, the dependence of refractive index on wavelength, variations in waveguide properties with wavelength, and transmission of two different polarizations of light through single-mode fiber. Like attenuation, dispersion can limit the distance a signal can travel through an optical fiber, but it does so in a different way. Dispersion does not weaken a signal; it blurs it. If you send one pulse every nanosecond but the pulses spread to 10 ns at the end of the fiber, they blur together. The signal is present, but it's so blurred in time that it is unintelligible. In its simplest sense, dispersion measures pulse spreading per unit distance in nanoseconds or picoseconds per kilometer. Total pulse spreading, t, is This equation actually gives dispersion in two different forms. One is the unit or characteristic dispersion of the fiber, written as dispersion and measured per unit length (in units of time per kilometer). The other is the total pulse spreading in units of time over the entire length. As long as the same fiber is used throughout the cable, the total pulse spreading is simply the characteristic fiber dispersion times the fiber length. If different types of fibers are used, you need to calculate pulse spreading separately for each section, then add them. The simple equation above holds for modal dispersion, which is the type most important for step-index multimode fibers, where modes travel at different speeds through the fiber. Graded-index fibers nominally equalize the speeds of all transmitted modes, but things don't work that perfectly in the real world. It's functionally impossible to achieve the ideal refractive-index profile needed to make all modes travel at exactly the same speed. That profile depends on wavelength, and fibers carry signals at a range of wavelengths. In practice, you have to rely on manufacturer specifications for the unit dispersion of graded-index fibers, typically specified in units of bandwidth (described below) rather than in time units. Other types of dispersion also add to total pulse spreading. We'll get to them in a minute, but first let's look at how to calculate the total pulse spreading. Material and waveguide dispersion add together to give a wavelength-dependent chromatic dispersion, mentioned in Chapter 4. Fibers also experience polarization-mode dispersion. Both quantities are independent of each other and of modal dispersion. That means you have to take the square root of the sum of the squares to get total pulse spreading: Polarization-mode dispersion doesn't matter in multimode fibers, so for that case the equation becomes Likewise, single-mode fibers have no modal dispersion (other than polarization-mode dispersion), so the equation becomes

Chromatic Dispersion and Wavelength

Chromatic dispersion is the pulse spreading that arises because the velocity of light through a fiber depends on its wavelength. It is measured in units of picoseconds (of pulse spreading) per nanometer (of spectral width of the optical signal) per kilometer (of fiber length). The total pulse spreading due to chromatic dispersion, tchromatic, is calculated by multiplying the fiber's characteristic chromatic dispersion by the range of wavelengths generated by the light source () and the fiber length:

The characteristic chromatic dispersion of a fiber is a function of wavelength. It is normally the largest type of dispersion in single-mode fiber systems. As you learned in Chapter 4, chromatic dispersion is the sum of two components, material and waveguide dispersion, which can cancel each other at certain wavelengths. In standard step-index single-mode fiber, material and waveguide dispersion add to zero near 1310 nm. Dispersion shifting moves the zero-dispersion point to other wavelengths, generally longer. To understand chromatic dispersion, we need to look at both material and waveguide dispersion.

Material dispersion arises from the change in a material's refractive index with wavelength. The higher the refractive index, the slower light travels. Thus as a pulse containing a range of wavelengths passes through a material, it stretches out, with the wavelengths with lower refractive index going faster than those with higher indexes. Like absorption, dispersion is a function of the individual material, which changes with wavelength. Communication fibers are nearly pure silica (SiO2), so their characteristic material dispersion is essentially the same as that of pure fused silica. Figure 5.7 plots both refractive index and material dispersion of fused silica against wavelength.

Note that material dispersion has a positive or negative sign, unlike the modal dispersion. You can think of this sign as indicating how the refractive index is changing with wavelength, although that's an oversimplification. The physical meaning of the sign is a bit obscure, but the signs are important in combining material dispersion and waveguide dispersion to calculate total chromatic dispersion. Although chromatic dispersion also has a sign, the calculations for total pulse spreading cancel it out because the formula uses the square of the pulse spreading caused by chromatic dispersion.

As Figure 5.7 shows, the magnitude of material dispersion is large at wavelengths shorter than 1.1 µm. High material dispersion at 850 nm makes chromatic dispersion high at that wavelength, limiting the transmission speed possible even in single-mode fiber. The real benefits of single-mode transmission come from operating at longer wavelengths where the material dispersion is small.

Waveguide dispersion is a separate effect, arising from the distribution of light between core and cladding. Recall that waveguide properties are a function of the wavelength. This means that changing the wavelength affects how light is guided in a single-mode fiber. For a step-index single-mode fiber, the waveguide dispersion is relatively small, but can be important. More complex refractive index profiles can increase waveguide dispersion, such as the dispersion-compensating fiber in Figure 4.12(f). Like material dispersion, waveguide dispersion has a sign that indicates how changing wavelength affects dispersion.

For most practical purposes, chromatic dispersion is the sum of material and waveguide dispersion.

Remember that the signs are important. From a physical standpoint, what happens is that the variation with wavelength caused by waveguide dispersion can offset (or add to) that caused by material dispersion. Dispersion shifting is done by designing fibers to have large negative waveguide dispersion, which offsets positive material dispersion at wavelengths longer than 1.28 µm, shifting the region of low chromatic dispersion near the erbium-fiber amplifier band. As mentioned earlier, generally the zero dispersion wavelength is chosen to be a little longer or shorter than the 1530- to 1620-n erbium-fiber band. Figure 5.8 shows how waveguide and material dispersion combine for step-index single-mode fiber and one type of nonzero dispersion-shifted fiber.

A Closer Look at Chromatic Dispersion

The descriptions of material, waveguide, and chromatic dispersion have been a bit vague because the formal definitions depend on some concepts that require a bit of extra work, a few equations, and a dash of calculus to understand. To delve more deeply, let's consider the case of material dispersion, which is the simplest because it depends only on how the refractive index of the material varies with wavelength. (Chromatic and waveguide dispersion work similarly, but the details are more complex.)

Recall that the velocity of light passing through a material depends on its refractive index. Since the refractive index varies with wavelength, so does the velocity of light in the material. Suppose that the material has a refractive index n1 at wavelength 1 and an index n2 at wavelength 2. The time each wavelength takes to pass through a length of glass L is

If you calculate the difference between the transit times at the two wavelengths, you get what is called the group delay time,

which measures the difference in travel time for the two wavelengths. This is the same as the pulse spreading through a fiber denoted by t.
From a physical standpoint, the group delay is the slope of the curve that plots refractive index as a function of wavelength, shown in Figure 5.9(a) on a different scale that shows its curvature better than Figure 5.7. If you know elementary calculus, that slope is the first derivative of how refractive index n varies with wavelength:

This group delay is plotted in Figure 5.9(b). You can think of group delay time as the actual pulse spreading t—masured in units of time—caused by the change in refractive index over a range of wavelengths. Remember, however, that this is a time delay, not the characteristic material dispersion of the fiber. Characteristic dispersion measures not the magnitude of the delay in units of time, but how fast the group delay is changing with wavelength
(generally for a unit length of the fiber rather than for the entire length). This characteristic material dispersion is measured in units of picoseconds (of time) per nanometer (of wavelength range) per kilometer (of fiber length). Multiply it by the length of the fiber and the range of wavelengths, and you get the group delay t.

You calculate characteristic material dispersion Dmaterial as the rate of change of the group delay with wavelength, which is equivalent to measuring the slope of the group delay curve with respect to wavelength. If you divide through by fiber length, and take the differential rate of change in group delay with wavelength, you get

This is the characteristic material dispersion, plotted in Figure 5.9(c), and it represents the slope of the group delay curve. To see what it means graphically, compare it with the plot of group delay in Figure 5.9(b). The group delay is nearly constant at its peak value, so the values are virtually the same at the two wavelengths near the peak (vertical lines). However, at shorter wavelengths the group delay is changing much faster, so the values differ much more at two wavelengths the same distance apart (vertical lines at the left).

You can calculate the total pulse spreading over the length of the fiber, t, by multiplying this characteristic dispersion by fiber length L and wavelength range . This gives:

Thus the characteristic material dispersion is proportional to the second derivative (or, equivalently, to the slope of the slope) of the plot of refractive index versus wavelength, not directly to the slope of the refractive index curve itself. To reiterate, it's also the slope of the group delay, which measures the travel time through the fiber as a function of wavelength. The slope of the group delay curve, in contrast, measures how fast the group delay changes with wavelength, which is the characteristic material dispersion. This rate of change of group delay is zero at the peak of the group delay curve, which comes at 1.28 µm in silica fibers. This also is the point where the slope of the refractive-index curve stops decreasing with increasing wavelength and starts increasing again. (Because the refractive index decreases as wavelength increases, the slope is a negative number, plotted below zero on Figure 5.9(b).) Mathematically, the zero material-dispersion wavelength is a maximum of the group velocity curve and a point of inflection in the refractive-index plot.

Figure 5.9(c) plots characteristic material dispersion. Recall that the formula carries a negative sign, which it gets from the negative value of group delay. The minus sign means that characteristic material dispersion is negative at wavelengths where the group delay curve is rising (i.e., has positive slope), and positive where the middle curve is dropping (i.e., has negative slope).

The components of waveguide dispersion work in a similar way, but the physical relationships are more complex. As you saw earlier, waveguide dispersion has a sign, which matters when adding it to material dispersion to get chromatic dispersion, the number given in product specifications. The sign also matters when compensating for chromatic dispersion to reduce pulse spreading. Chromatic dispersion works like material dispersion; it measures the rate of change of the group delay for all chromatic dispersion, not just for material dispersion.

The signs don't matter when combining the effects of chromatic dispersion with other dispersion, because the pulse spreading enters those equations as squares. As you've probably learned the hard way, it's easy to lose track of signs that don't have an obvious physical meaning. This can happen very easily with material, waveguide, and chromatic dispersion, so don't be surprised if you spot the wrong signs. In normal single-mode fibers, the dispersion should be negative at wavelengths shorter than the zero-dispersion wavelength, and positive at longer wavelengths.

Dispersion Slope and Specifications

In practice, engineers approximate chromatic dispersion by assuming it varies linearly over a defined range of wavelengths. That is, they plot dispersion at a pair of wavelengths, draw a straight line between them, and assume that the dispersion at intermediate wavelengths falls between them, as shown in Figure 5.10. If the two wavelengths are 1 and 2, and the characteristic chromatic dispersions at those wavelengths are D(1) and D(2), this means that dispersion Dchromatic at intermediate wavelength is

Specification sheets often give these equations with the ranges of dispersion and wavelength for which they are valid. Typically there are separate equations for the 1530- to 1565-nm range of C-band erbium-doped fiber amplifiers and the 1565- to 1625-nm L-band.

Look closely at the equation, and you can see that it actually multiplies the dispersion slope (change in dispersion over a range of wavelength) by the change in wavelength from one endpoint, and adds the dispersion at that endpoint. Thus the equation translates in more descriptive terms to

Remember this is the slope of chromatic dispersion, although the normal term is just "dispersion slope."

Dispersion slope tells how dispersion changes with wavelength. Normally this change is very small over the range of wavelengths generated by a single laser transmitter. However, it is important in wavelength-division multiplexed systems, which carry many optical channels spanning tens of nanometers in wavelength. We will take a closer look later in this chapter.

Specification sheets typically do not plot chromatic dispersion directly as a function of wavelength, but give the chromatic dispersion that may be found at a range of wavelengths, such as 2.6 to 6.0 ps/nm-km at 1530 to 1565 nm. These numbers do not mean that the fibers

have 2.6 ps/nm-km dispersion at 1530 nm and 6.0 ps/nm-km at 1565—they mean that the values in this range of wavelengths fall within this "box." As with other specified values, they allow for a range of manufacturing tolerances, so the specified dispersion slope will not al ways match the slope calculated from the extremes of chromatic dispersion and wavelength

Source Bandwidth and Chromatic Dispersion

Unlike the pulse spreading caused by other types of fiber dispersion, the spreading caused by chromatic dispersion depends strongly on the light source. If we take Dchromatic () as the characteristic dispersion of a unit length (1 km) of fiber, the total pulse spreading from chromatic dispersion tchromatic is

where is the range of wavelengths in the optical signal in nanometers and Length is the fiber length in kilometers. This means that the spectral bandwidth of the light source is a parameter that system designers can adjust to limit the effects of chromatic dispersion. The higher the data rate, the more important narrow-band sources become, as you will learn in later chapters.

Chromatic Dispersion Compensation and Tailoring

We saw before that pulse dispersion is cumulative, adding up over the entire length of a fiber system. In general that means that adding more fiber only makes things worse. However, it is possible to cancel chromatic dispersion if you add fiber with chromatic dispersion of the opposite sign. For example, if you have a fiber with positive chromatic dispersion in the erbium-amplifier band, you can add a fiber with negative chromatic dispersion at the same waveIengths. The idea is similar to having waveguide dispersion offset material dispersion to give low chromatic dispersion at the desired wavelength. In this case, it is two separate fibers, spliced together, that offset each other's dispersion, as shown in Figure 5.11. Dispersion compensation modules can be added to existing fiber systems to upgrade them for higher-speed transmission or operation in the 1550-nm band, or fiber with different dispersion can be integrated into the transmission paths of new systems to "tailor" overall dispersion properties.

A typical dispersion-compensating fiber has high negative waveguide dispersion that gives it a negative chromatic dispersion that typically is several times the magnitude of the positive chromatic dispersion of the transmission fiber. Thus compensation requires a shorter length of the dispersion-compensating fiber. That is an important consideration because compensating fiber typically has higher attenuation than transmission fibers. Compensating fiber also usually has a small effective area, making it more vulnerable to nonlinear effects, so it is used at the receiving end of the system, where lower power reduces nonlinear effects.

Typically, dispersion is compensated over a range of wavelengths in the erbium-fiber band, but it's easiest to calculate requirements if you look just at one wavelength. Suppose you want to have total chromatic dispersion of +2 ps/nm-km at the 1530-nm short end of the erbium-fiber band over a 1000-km system. You are using nonzero dispersion-shifted transmission fiber with dispersion of +8 ps/nm-km at that wavelength, and you can buy dis-

persion-compensating fiber with dispersion of –100 ps/nm-km at 1530 nm. You can use the general formula

Where Dnet, is the net dispersion for the entire system, Ltotal is the total length (assuming the compensating fiber is part of the transmission path), Dtransmission and Ltransmission are the dispersion and length of the transmission fiber, and Dcomp and Lcom are dispersion and length of the compensating fiber. Plug the numbers in, and you see

Since you know that Ltransmission + Lcomp = 1000 km, you can work out that you need 944 km of nonzero dispersion-shifted transmission fiber and 56 km of compensating fiber. Thus you need about 1 km of compensating fiber for every 17 km of transmission fiber.

You can use the same ideas to calculate the dispersion compensation needed for upgrading existing fiber systems. Other approaches to chromatic dispersion also are possible. One example is an optical delay line that would delay signals a certain amount depending on their wavelength, so the slower signals could catch up. A dispersion-compensating fiber in a box could serve as such a delay line.

Multiwavelength Transmission and Dispersion

Dealing with chromatic dispersion is more complex in systems that carry multiple wavelengths. Wavelength-division multiplexing requires management of chromatic dispersion over the entire range of wavelengths that are transmitting optical channels. Typically that can span tens of nanometers in systems with fiber amplifiers, 35 nm in systems with C-band erbium-fiber amplifiers, 55 nm in systems with L-band erbium-fiber amplifiers, or 95 nm in systems with both.

That range of wavelength is large enough for chromatic dispersion to differ significantly among optical channels. Just in the erbium-fiber C-band, the difference can accumulate to 2 ps/nm-km with reduced-dispersion-slope (0.045 ps/nm2-km) fibers, and to 4 ps/nm-km with other nonzero dispersion-shifted fibers. This becomes important because it means different optical channels may require different amounts of dispersion compensation.

Dispersion management also becomes more complex as the range of wavelengths increases. The dispersion slopes of dispersion-compensating fibers do nor match and offset those of transmission fibers, so residual differences remain. These accumulate over distance and can become significant for long-distance, high-speed systems. Additional components or a mix of dispersion-compensating and transmission fibers may be needed.

Polarization-Mode Dispersion

In Chapter 4, you learned that a single-mode fiber actually transmits light in two distinct polarization modes. The electric fields of the two modes are perpendicular to each other, or orthogonal in the jargon of physics. Normally the two behave just the same in the fiber, so from a physical standpoint they are called degenerate, which means they can't be distinguished.

The existence of two polarization modes wouldn't matter if they were perfectly identical, but in the real world even physics is not quite perfect. The polarization modes behave identically only in a fiber that is perfectly symmetrical, and no fiber is that perfect. Stresses within the fiber, and forces applied to it from the outside world, cause the refractive index of glass to differ very slightly for light in the two polarization modes, an effect called birefringence.

Internal forces make some crystals strongly birefringent; calcite is one example, where the refractive indexes of the two polarizations differ by more than 10%. Look through a calcite prism and you see what looks like a double exposure because the material separates light in the two polarizations. The effect in optical fibers is very tiny—around one part in 10 million (10–7) from manufacturing stresses—but light travels a very long distance in them, so tiny effects can accumulate. If the birefringence was uniform along the whole fiber, light in one polarization mode would get about one wavelength farther ahead of the other every 10 meters, as shown in Figure 5.12. However, the effect called polarization-mode dispersion (PMD) isn't that simple.

One reason is that birefringence varies randomly along the length of normal single-mode fibers, because it arises from random fluctuations in manufacturing and environmental stresses distributed along the fiber's length. Another is that the light can shift randomly between polarization modes in normal single-mode fibers (but not in polarization-maintaining or single-polarization fibers, mentioned in Chapter 4). That means that the phase shift

between the two modes does not accumulate consistently along the length of the fiber. Instead, the light drifts back and forth between the two modes, and the difference blurs, stretching the pulse duration in time, as you can see at the bottom of Figure 5.12.

The random polarization shifts and distribution of birefringence cause the pulse spreading to increase with the square root of fiber length L. The effect is sometimes called differential group delay, and is essentially statistical. Each fiber has a characteristic polarization-mode dispersion DPMD when manufactured, but that may change when cabled. Pulse spreading caused by polarization-mode dispersion is tPMD is

Typical values of existing fiber are 0.05 to 1 picosecond per root kilometer. Fiber now in production has characteristic PMD values of 0.05 to 0.2 ps/km–1/2 but cabling can raise the value to 0.5 ps/km–1/2 The value may change with time, such as when overhead cables are strained by blowing in the wind, which can increase the instantaneous values of characteristic PMD to over 1 ps/km–1/2. The magnitude also can vary with wavelength.

The potential effects of polarization mode dispersion were not considered significant until a few years ago, so manufacturers did not specify PMD values for their earlier fibers. Because cabling effects also are important, the best way to be sure of PMD values in existing cables is to measure them in place. In principle, all optical components can cause polarization mode dispersion, but in practice fiber is the most important source because of the long optical path length.

Polarization-mode dispersion is smaller in magnitude than chromatic dispersion, but technology to compensate for it is still in the early stages of development. In practice, it does not become significant until data rates exceed 2.5 Gbit/s. Careful control allows long-distance transmission at 10 Gbit/s, but PMD poses a challenge to sending higher data rates over long distances.

Dispersion and Transmission Speeds

So far we have considered the effects of dispersion on instantaneous pulses, but real pulses are not instantaneous. In digital systems, the initial pulse starts with a duration tinput then experiences spreading due to dispersion of tdispersion. The output pulse length is not the direct sum of the two pulse durations but the root sum of their squares:

This gives the pulse length at the end of the system, and it is these pulses that have to be resolved for the system to operate properly.

The degree of overlap at which pulse dispersion causes problems in digital systems depends on the design. One rough guideline for estimating the maximum bit rate is that the interval between pulses should be four times the dispersion, or, equivalently,

Thus, if pulses experience about 1 ns of dispersion, the maximum bit rate is about 250 Mbit/s. It isn't quite this simple in practice because performance depends on other factors as well as dispersion, but it's a useful guideline. For polarization-mode dispersion the usual guideline is more stringent, that the dispersed pulse should be no more than 1/10th as long as the interval between pulses, reflecting the more stringent demands on high-speed systems. Note that these figures consider only dispersion, not the input pulse length, or receiver rise time. Different guidelines relate total system rise time to maximum bit rate, which depend on data transmission format.

Dispersion also affects analog transmission in roughly the same way that it limits bit rates in digital systems. Instead of lengthening digital pulses, dispersion smears out the whole analog waveform, effectively attenuating the highest frequencies in the signal. This limits the analog bandwidth, the frequency at which the detectable signal has dropped 3 dB (50%) compared to lower frequencies.

Transmission capacities of graded-index and step-index multimode fiber often are specified in terms of bandwidth, typically megahertz-kilometers, rather than as dispersion. You can roughly convert that to total system response time ttotal using the formula

Typical bandwidths for step-index multimode fiber are around 20 MHz-km. Bandwidths of graded-index multimode fibers depend on wavelength because they suffer both chromatic and modal dispersion. Typical values at 0.85 µm are 200 MHz-km for 62.5/125 fiber and 600 MHz-km for 50/125 fiber. At 1.3 µm, typical bandwidths are 500 MHz/km for 62.5/125 fiber and 1000 MHz/km for 50/125 fiber. Because multimode graded-index fibers are used in local area networks, they also may be specified according to maximum transmission distance using particular Ethernet protocols.

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