320 TRANSMISSION SYSTEM ENGINEERING 5.7.4 Polarization-Mode Dispersion (PMD) The origin of PMD lies in the fact that different polarizations travel with different group velocities because of the ellipticity of the fiber core; we discussed this in Section 2.1.2. Moreover, the distribution of signal energy over the different state of polarizations (SOPs) changes slowly with time, for example, because of changes in the ambient temperature. This causes the PMD penalty to vary with time as well. In addition to the fiber itself, PMD can arise from individual components used in the network. The time-averaged differential time delay between the two orthogonal SOPs on a link is known to obey the relation [KK97a, Chapter 6] <AT)- DPMD~/L, where (At) is called the differential group delay (DGD), L is the link length, and DPMD is the fiber PMD parameter, measured in ps/kx/-kmm. The PMD for typical fiber lies between 0.5 and 2 ps/kx/~m. However, carefully constructed new links can have PMD as low as 0.1 ps/k~/k-m. In reality, the SOPs vary slowly with time, and the actual DGD Ar is a random variable. It is commonly assumed to have a Maxwellian probability density function (see Appendix H). This means that the square of the DGD is modeled by a more familiar distributionmthe exponential distribution. The larger the DGD, the larger is the power penalty due to PMD. Thus, the power penalty due to PMD is also time varying, and it turns out that it is proportional to Ar 2 and thus obeys an exponential distribution (see Problem 5.22). If the power penalty due to PMD is large, it is termed a PMD outage and the link has effectively failed. For a DGD of 0.3T, where T is the bit duration, the power penalty is approximately 0.5 dB for a receiver limited by thermal noise and 1 dB for a receiver with signal-dependent noise (ITU G.691). Using the Maxwellian distribution, the probability that the actual delay will be greater than three times the average delay is about 4 x 10 -5 (see Appendix H). Given our earlier reasoning, this means that in order to restrict the PMD outage probability (PMD >_ 1 dB) to 4 x 10 -5, we must have the average DGD to be less than 0.1T; that is, (A'c) DPMDX/~ < 0.1T. (5.23) This limit is plotted in Figure 5.24. Observe that for a bad fiber with PMD of 2 ps/k,,/-k-~m, the limit is only 25 km. This is an extreme case, but it points out that PMD can impose a significant limitation. Note that we have not said anything about the distribution of the length of time for which there is a PMD outage. In the above example, the DGD may exceed three times the average delay, and we may have one PMD outage with an average duration 5.7 Dispersion 321 r~ ,.0 4 , . , ~ 1000 500 100 50 10 5 .1 DpMD=2"0 ~ ~ , , , ,,,ll 10 100 1000 10,000 Distance, L (kin) Figure 5.24 Limitations on the simultaneously achievable bit rates and distances im- posed by PMD. of one day once every 70 years, or one with an average duration of one minute every 17 days. This depends on the fiber cable in question, and typical outages last for a few minutes. Thus an outage probability of 4 x 10 -5 can also be interpreted as a cumulative outage of about 20 minutes per year. The limitations due to intermodal dispersion, chromatic dispersion, and PMD are compared in Figure 5.25. PMD gives rise to intersymbol interference (ISI) due to pulse spreading, just as all other forms of dispersion. The traditional (electronic) technique for overcoming ISI in digital systems is equalization, discussed in Section 4.4.9. Equalization to compensate for PMD can be carried out in the electronic domain and is discussed in [WK92]. However, electronic equalization becomes more difficult as the bit rate increases and, to date, is not feasible for 40 Gb/s systems. At such high bit rates, optical PMD compensation must be used. To understand how PMD can be compensated optically, recall that PMD arises due to the fiber birefringence and is illustrated in Figure 2.5. The transmitted pulse consists of a "fast" and a "slow" polarization component. The principle of PMD compensation is to split the received signal into its fast and slow polarization compo- nents, and to delay the fast component so that the DGD between the two components is compensated. Since the DGD varies in time, the delay that must be introduced in the fast component to compensate for PMD must be estimated in real time from the properties of the link. 322 TRANSMISSION SYSTEM ENGINEERING 1000 - 500 100 ,,.0 ~ 5o N lO 5 Chromatic (0 nm) "~omatic (1 nm) ~ ~M~ a ~, , ,,, ~ ,,~ 10 100 1000 10,000 Distance, L (km) Figure 5.25 Limitations on the simultaneously achievable bit rates and distances im- posed by intermodal dispersion, chromatic dispersion with a source spectral width of 1 nm, chromatic dispersion with spectral width proportional to the modulation band- width, and PMD with DpMD = 0.5. NRZ modulation transmission over standard single-mode fiber with a chromatic dispersion value of 17 ps/nm-km is assumed. The PMD effect we have discussed so far must strictly be called first-order polarization-mode dispersion. First-order PMD is a consequence of the fact that the two orthogonal polarization modes in optical fiber travel at slightly different speeds, which leads to a differential time delay between these two modes. However, this differential time delay itself is frequency dependent and varies over the band- width of the transmitted pulse. This effect is called second-order PMD. Second-order PMD is an effect that is similar to chromatic dispersion and thus can lead to pulse spreading. PMD also depends on whether RZ or NRZ modulation is used; the discussion so far pertains to NRZ modulation. For RZ modulation, the use of short pulses enables more PMD to be tolerated since the output pulse has more room to spread similar to the case of chromatic dispersion. However, second-order PMD depends on the spectral width of the pulse; narrower pulses have larger spectral widths. This is similar to the case of chromatic dispersion (Section 5.7.2). Again, as in the case of chromatic dispersion, there is an optimum input pulse width for RZ modulation that minimizes the output pulse width [SKA00, SKA01]. In addition to PMD, some other polarization-dependent effects influence sys- tem performance. One of these arises from the fact that many components have a polarization-dependent loss (PDL); that is, the loss through the component depends 5.8 Fiber Nonlinearities 323 on the state of polarization. These losses accumulate in a system with many compo- nents in the transmission path. Again, since the state of polarization fluctuates with time, the signal-to-noise ratio at the end of the path will also fluctuate with time, and careful attention needs to be paid to maintain the total PDL on the path to within acceptable limits. An example of this is a simple angled-facet connector used in some systems to reduce reflections. This connector can have a PDL of about 0.1 dB, but hundreds of such connectors can be present in the transmission path. 5.8 Fiber Nonlinearities As long as the optical power within an optical fiber is small, the fiber can be treated as a linear medium; that is, the loss and refractive index of the fiber are independent of the signal power. However, when power levels get fairly high in the system, we have to worry about the impact of nonlinear effects, which arise because, in reality, both the loss (gain) and refractive index depend on the optical power in the fiber. Nonlinearities can place significant limitations on high-speed systems as well as WDM systems. As discussed in Chapter 2, nonlinearities can be classified into two categories. The first occurs because of scattering effects in the fiber medium due to the interaction of light waves with phonons (molecular vibrations) in the silica medium. The two main effects in this category are stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS). The second set of effects occurs because of the dependence of refractive index on the optical power. This category includes four-wave mix- ing (FWM), self-phase modulation (SPM), and cross-phase modulation (CPM). In Chapter 2, we looked at the origins of all these effects. Here we will understand the limitations that all these nonlinearities place on system designers. Except for SPM and CPM, all these effects provide gains to some channels at the expense of depleting power from other channels. SPM and CPM, on the other hand, affect only the phase of signals and can cause spectral broadening, which in turn, leads to increased chromatic dispersion penalties. 5.8.1 Effective Length in Amplified Systems We discussed the notion of the effective length of a fiber span in Section 2.4.1. In systems with optical amplifiers, the signal gets amplified at each amplifier stage without resetting the effects due to nonlinearities from the previous span. Thus the effective length in such a system is the sum of the effective lengths of each span. 324 TRANSMISSION SYSTEM ENGINEERING 10,000 " 1000 loo 10 J lO l =~km .,/'~namplified system f I I i i i ii II i i i i i I III i i i i i i.ii 100 1000 10,000 Link length (km) Figure 5.26 Effective transmission length as a function of link length, 1. In a link of length L with amplifiers spaced 1 km apart, the effective length is approximately given by 1 -e -~I L Le = (5.24) ot / ,, Figure 5.26 shows the effective length plotted against the actual length of the trans- mission link for unamplified and amplified systems. The figure indicates that, in order to reduce the effective length, it is better to have fewer amplifiers spaced fur- ther apart. However, what matters in terms of the system effects of nonlinearities is not just the effective length; it is the product of the launched power P and the effective length Le. Figure 5.6 showed how P varies with the amplifier spacing 1. Now we are interested in finding out how P Le grows with the amplifier spacing 1. This is shown in Figure 5.27. The figure shows that the effect of nonlinearities can be reduced by reducing the amplifier spacing. Although this may make it easier to design the amplifiers (they need lower gain), we will also need more amplifiers, resulting in an increase in system cost. The effect of a scattering nonlinearity depends on P Le and thus increases with an increase in the input power and the link length. The longer the link, the greater is the amount of power that is coupled out from the signal (pump) into the Stokes wave. For a given link length, an approximate measure of the power level at which the effect of a nonlinearity starts becoming significant is the threshold power. For a given fiber length, the threshold power of a scattering nonlinearity is defined as the incident optical power per channel into the fiber at which the pump and Stokes 5.8 Fiber Nonlinearities 325 CD r~ ~D 10,000 1000 100 10 | , | 0 50 100 150 200 250 300 Amplifier spacing (km) Figure 5.27 Relative value of PLe versus amplifier spacing. The ordinate is the value relative to an amplifier spacing of 1 km. a - 0.22 dB/km. powers at the fiber output are equal. In amplified systems, the threshold power is reduced because of the increase in the effective length. This makes amplified systems more susceptible to impairments due to nonlinearities. 5.8.2 Stimulated Brillouin Scattering The calculation of the threshold power for SBS Pth is quite involved, and we simply state the following approximation for it from [Smi72]" Pth 21bAe gBLe Here, Ae and Le are the effective area and length of the fiber, respectively (see Section 2.4.1), gB ~ 4 x 10 -11 m/W is called the Brillouin gain coefficient, and the value of b lies between 1 and 2 depending on the relative polarizations of the pump and Stokes waves. Assuming the worst-case value of b = 1, Ae 50 #m 2, and Le -~ 20 km, we get Pth = 1.3 mW. Since this is a low value, some care must be taken in the design of optical communication systems to reduce the SBS penalty. The preceding expression assumes that the pump signal has a very narrow spectral width and lies within the narrow 20 MHz gain bandwidth of SBS. The threshold power is considerably increased if the signal has a broad spectral width, and thus 326 TRANSMISSION SYSTEM ENGINEERING 5.8.3 much of the pump power lies outside the 20 MHz gain bandwidth of SBS. An approximate expression that incorporates this effect is given by ( source) eth ~ 1+ gBLe AfB ' where Afsourc e is the spectral width of the source. With Afsourc e = 200 MHz, and still assuming b - 1, the SBS threshold increases to Pth = 14.4 mW. The SBS penalty can be reduced in several ways" 1. Keep the power per channel to much below the SBS threshold. The trade-off is that in a long-haul system, we may have to reduce the amplifier spacing. 2. Since the gain bandwidth of SBS is very small, its effect can be decreased by increasing the spectral width of the source. This can be done by directly modu- lating the laser, which causes the spectral width to increase because of chirp. This may cause a significant chromatic dispersion penalty. The chromatic dispersion penalty can, however, be reduced by suitable chromatic dispersion management, as we will see later. Another approach is to dither the laser slightly in frequency, say, at 200 MHz, which does not cause as high a penalty because of chromatic dispersion but increases the SBS threshold power by an order of magnitude, as we saw earlier. This approach is commonly employed in high-bit-rate systems transmitting at high powers. Irrespective of the bit rate, the use of an external modulator along with a narrow spectral width source increases the SBS threshold by only a small factor (between 2 and 4) for amplitude-modulated systems. This is because a good fraction of the power is still contained in the optical carrier for such systems. 3. Use phase modulation schemes rather than amplitude modulation schemes. This reduces the power present in the optical carrier, thus reducing the SBS penalty. In this case, the spectral width of the source can be taken to be proportional to the bit rate. However, this may not be a practical option in most systems. Stimulated Raman Scattering We saw in Section 2.4 that if two or more signals at different wavelengths are injected into a fiber, SRS causes power to be transferred from the shorter-wavelength chan- nels to the longer-wavelength channels (see Figure 2.16). Channels up to 150 THz (125 nm) apart are coupled due to SRS, with the peak coupling occurring at a separation of about 13 THz. Coupling occurs for both copropagating and counter- propagating waves. Coupling occurs between two channels only if both channels are sending 1 bits (that is, power is present in both channels). Thus the SRS penalty is reduced when 5.8 Fiber Nonlinearities 327 chromatic dispersion is present because the signals in the different channels travel at different velocities, reducing the probability of overlap between pulses at different wavelengths at any point in the fiber. This is the same pulse walk-off phenomenon that we discussed in the case of CPM in Section 2.4.7. Typically, chromatic dispersion reduces the SRS effect by a factor of 2. To calculate the effect of SRS in a multichannel system, following [Chr84], we approximate the Raman gain shape as a triangle, where the Raman gain coefficient as a function of wavelength spacing A)~ is given by /xz if 0 < A)~ < A)~c, gRS~c, g (A)~) = 0 otherwise. Here A)~c = 125 nm, and gR ~ 6 x 10 -14 ~ (at 1.55/zm) is the peak Raman gain coefficient. Consider a system with W equally spaced channels 0, 1 W- 1, with A~s denoting the channel spacing. Assume that all the channels fall within the Raman gain bandwidth; that is, the system bandwidth A = (W- 1)A;~s < A)~c. This is the case of practical interest given that the Raman gain bandwidth is 125 nm and the channels within a WDM system must usually be spaced within a 30 nm band dictated by the bandwidth of optical amplifiers. The worst affected channel is the channel corresponding to the lowest wavelength, channel 0, when there is a 1 bit in all the channels. Assume that the transmitted power is the same on all channels. Assume further that there is no interaction between the other channels, and the powers of the other channels remain the same (this approximation yields very small estimation errors). Assume also that the polarizations are scrambled. This is the case in practical systems. In systems that use polarization-maintaining fiber, the Raman interaction is enhanced, and the equation that follows does not have the factor of 2 in the denominator. The fraction of the power coupled from the worst affected channel, channel 0, to channel i is given approximately by [Buc95] Po(i) gR i A)~s P Le A)~c 2Ae This expression can be derived starting from the coupled wave equations for SRS that are similar in form to (2.14) and (2.15); see [Buc95] for details and [Zir98] for an alternate derivation with fewer assumptions. So the fraction of the power coupled out of channel 0 to all the other channels is W-1 Po ~ Po(i)= i=1 gRA)~sPLe W(W- 1) 2 A )~c Ae 2 (5.25) 328 TRANSMISSION SYSTEM ENGINEERING 40 ~n ~D ~ 3o ~ 20 ~ 10 0 ~ 0 -10 | I I I ' | I1111 -20100 200 ,, , 500 1000 2000 5000 10,000 Link length (km) Figure 5.28 Limitation on the maximum transmit power per channel imposed by stim- ulated Raman scattering. The channel spacing is assumed to be 0.8 nm, and amplifiers are assumed to be spaced 80 km apart. The power penalty for this channel is then - 10 log(1 - Po). In order to keep the penalty below 0.5 dB, we must have Po < 0.1, or, from (5.25), WP(W- 1)AXsLe < 40,000 mW-nm-km. Observe that the total system bandwidth is A = (W- 1)A)~s and the total transmitted power is Ptot - WP. Thus the result can be restated as PtotALe < 40,000 mW-nm-km. The preceding formula was derived assuming that no chromatic dispersion is present in the system. With chromatic dispersion present, the right-hand side can be relaxed to approximately 80,000 mW-nm-km. If the channel spacing is fixed, the power that can be launched decreases with W as 1/W 2. For example, in a 32-wavelength system with channels spaced 0.8 nm (100 GHz) apart, and Le = 20 km, P _< 2.5 mW. Figure 5.28 plots the maximum allowed transmit power per channel as a function of the link length.The limit plotted here corresponds to PtotALe < 80,000 mW-nm-km. Although SRS is not a significant problem in systems with a small number of channels due to the relatively high threshold power, it can pose a serious problem 5.8 Fiber Nonlinearities 329 0)1 0)3 0)2 0)113 0)213 0)223 0)132 0)221 0)231 0)331 0)123 0)312 0)321 0)112 0)332 Figure 5.29 Four-wave mixing terms caused by the beating of three equally spaced channels at frequencies o)1, o)2, and COB. in systems with a large number of wavelengths. To alleviate the effects of SRS, we can (1) keep the channels spaced as closely together as possible, and/or (2) keep the power levels below the threshold, which will require us to reduce the distance between amplifiers. 5.8.4 Four-Wave Mixing We saw in Section 2.4 that the nonlinear polarization causes three signals at frequen- cies o)i, o)j, and O)k to interact to produce signals at frequencies wi + ~oj + o)~. Among these signals, the most troublesome one is the signal corresponding to O)ij k o)i + O)j (-Ok, i ~: k, j ~: k. (5.26) Depending on the individual frequencies, this beat signal may lie on or very close to one of the individual channels in frequency, resulting in significant crosstalk to that channel. In a multichannel system with W channels, this effect results in a large number (W(W - 1) 2) of interfering signals corresponding to i, j, k varying from 1 to W in (5.26). In a system with three channels, for example, 12 interfering terms are produced, as shown in Figure 5.29. Interestingly, the effect of four-wave mixing depends on the phase relationship between the interacting signals. If all the interfering signals travel with the same group velocity, as would be the case if there were no chromatic dispersion, the effect is reinforced. On the other hand, with chromatic dispersion present, the different signals travel with different group velocities. Thus the different waves alternately overlap in and out of phase, and the net effect is to reduce the mixing efficiency. The . equally spaced channels 0, 1 W- 1, with A~ s denoting the channel spacing. Assume that all the channels fall within the Raman gain bandwidth; that is, the system bandwidth A = (W- 1 )A; ~s < A) ~c estimation errors). Assume also that the polarizations are scrambled. This is the case in practical systems. In systems that use polarization-maintaining fiber, the Raman interaction is enhanced,. Again, since the state of polarization fluctuates with time, the signal-to-noise ratio at the end of the path will also fluctuate with time, and careful attention needs to be paid to maintain