Dispersion Compensation for High Bit Rate WDM Channels
Commercial WDM systems in which individual channels operate at a bit rate of 40 Gb/s became available by 2002, and efforts were underway to increase the channel bit rate to 100 Gb/s and beyond. For such high-speed systems, the management of channel dispersion poses additional problems. In this tutorial, we focus on several relevant issues.
1. Tunable Dispersion Compensation
It is difficult to attain full GVD compensation for all channels in a WDM system. A small amount of residual dispersion remains uncompensated and often becomes a concern for long-haul systems. For a link of length L, this accumulated dispersion is given by
where D(z) denotes local dispersion along the link. A post-compensation technique is often adopted in laboratory experiments. In this approach, the residual dispersion for individual channels is compensated by adding adjustable lengths of a dispersion-compensating fiber (DCF) at the receiver end (dispersion trimming). This technique is not suitable for commercial WDM systems for several reasons. First, the exact amount of channel-dependent residual dispersion is not always known because of uncontrollable variations in the dispersion of fiber segments forming the transmission path. Second, even the path length may change in reconfigurable optical networks. Third, as the single-channel bit rate increases toward 40 Gb/s and beyond, the tolerable value of the residual dispersion becomes so small that even temperature-induced changes in GVD become a concern. For these reasons, the best approach is to adopt a tunable dispersion-compensation scheme that allows the dispersion control for each channel in a dynamic fashion.
Many techniques have been developed for tunable dispersion compensation over the last decade. Several of them make use of a fiber Bragg grating whose dispersion is tuned by changing the optical period 
of the grating. In one scheme, the grating is chirped nonlinearly so that its Bragg wavelength increases nonlinearly along the grating length. Tunable dispersion is realized when such a grating is stretched with a piezoelectric transducer.
In a linearly chirped grating, the slope of group delay (responsible for dispersion) at a given wavelength does not change with stretching. However, this slope can be changed by a large factor when the chirp is nonlinear. Mathematically, stress-induced changes in the mode index 
change the local Bragg wavelength as
For such a grating, the grating dispersion becomes
where τg is the group delay and Lg is the grating length. The value of Dg at any wavelength can be altered by changing the mode index (through heating or stretching), resulting in tunable dispersion characteristics for the Bragg grating.
The stretching technique has been used with success since 1999 to tune the dispersion provided by a nonlinearly chirped fiber grating. The grating is placed on a mechanical stretcher and a piezoelectric transducer is used to stretch it by applying an external voltage. The figure below shows the group-delay characteristics of a 5-cm-long grating as the voltage is changed from 0 to 1,000 V. For a fixed channel wavelength, da can be changed from -300 to -1,000 ps/nm by changing the voltage, resulting in a tuning range of 700 pcs/nm. The same technique can be extended to provide tunable compensation for multiple channels by using a sampled grating with nonlinear chirp. However, it suffers from a relatively large third-order dispersion that affects each channel. This problem can be solved by cascading two identical gratings in a fashion such that their chirps are opposite in nature.
In a different approach to realizing tunable dispersion, the grating is made with either no chirp or with a linear chirp, and a temperature gradient is used to produce a controllable chirp. Such distributed heating requires a thin-film heater deposited on the outer surface of the fiber whose core contains the grating. In a simple technique, film thickness is changed along the grating length to create a temperature gradient when a constant voltage is applied across the film. Figure (a) below shows the reflection spectra of a 8-cm-long grating at three voltage levels. The total dispersion, calculated from the group delay τg(λ), is displayed in figure (b) as a function of applied voltage. The grating is initially unchirped and has a narrow stop band that shifts and broadens as the grating is chirped through nonuniform heating. Physically, the Bragg wavelength λB changes along the grating because the optical period 
becomes z-dependent when a temperature gradient is established along the grating. The total dispersion DgLg can be changed in the range of -500 to -2,200 ps/nm by this approach. Such gratings can be used to provide tunable dispersion for 10-Gb/s systems. A segmented thin-film heater is sometimes used for creating a temperature gradient as it provides better temperature control along the grating length. Both the dispersion and dispersion slope of such a device can be controlled electronically. Moreover, in contrast to the stretching technique that requires large voltages, only a few volts are required for thermal tuning of a grating.
Cascaded, phase-apodized, chirped fiber gratings can also provide tunable dispersion with thermal tuning. The figure below shows such a device schematically. It consists in two such gratings cascaded using a four-port optical circulator. Each phase-apodized grating consists of two superimposed gratings of different periods and thus acts as a distributed GT filter (refer to this dispersion-equalizing filter tutorial about GT filter) with a group delay that varies periodically with frequency with a period equal to the free spectral range of the GT filter. Dispersion tuning is realized with the help of multiple heating elements along the length of each grating that are used to change the local grating period. By adjusting the group delays in the two gratings with suitable temperature profiles, such a device can compensate simultaneously the dispersion of 32 channels, spaced apart by 50 GHz, while providing a tuning range of ±800 ps/nm over a 30-GHz bandwidth.
As seen in the dispersion-equalizing filter tutorial, planar lightwave circuits, fabricated with the silica-on-silicon technology, can be used as tunable dispersion compensators. The use of an arrayed waveguide grating (AWG) based on this technology provides another approach to realizing tunable dispersion. The figure below shows such a device schematically. It consists of an AWG that is attached to a polymer-based planar lightwave circuit containing a thermo-optic lens. The AWG has a free spectral range of 100 GHz so that it can demultiplex a WDM signal containing 100-GHz-spaced channels. The polymer PLC contains a 7.5-μm-thick slab waveguide with an array of 16 heaters on top that can be individually addressed to produce a parabolic heat distribution. A mirror at the far end of the 4.2-mm-long polymer waveguide reflects all channels back toward the AWG. The dispersion of such a device could be tuned over a 1300-ps/nm range over a 40-GHz bandwidth.
In another AWG-based approach shown in the figure below, the demultiplexed channels are focused onto an array of liquid-crystal elements that reflect each channel back after imposing an electrically controllable phase shift on it. Such a device was used to compensate dispersion of a WDM signal occupying the entire L band. In a 2009 experiment, the liquid-crystal array was replaced with lens-shaped trenches filled with optical resins.
2. Higher-Order Dispersion Management
When the bit rate of a single channel exceeds 40 Gb/s (through the use of time-division multiplexing, e.g.), the third- and higher-order dispersive effects begin to influence the optical signal. For example, the bit slot at a bit rate of 160 Gb/s is only 6.25 ps wide. An RZ optical signal at such a high bit rate consists of pulses of width <5 ps. The following equation can be used to estimate the maximum transmission distance L, limited by the third-order dispersion (TOD) β3, when the second-order dispersion (GVD) is fully compensated (refer to this dispersion-induced limitations tutorial).
The result of L is found to be
This limitation is shown in the following figure by the dashed line.
At a bit rate of 200 Gb/s, L is limited to about 50 km and drops to only 3.4 km at 500 Gb/s if we use a typical value of β3 = 0.08 ps3/km. Clearly, it is essential to develop devices that can compensate for both the GVD and TOD in a tunable fashion when the single-channel bit rate exceeds 100 Gb/s.
The simplest solution to TOD compensation is provided by dispersion-compensating fibers (DCFs) designed to have a negative dispersion slope so that both β2 and β3 have opposite signs in comparison with the standard fibers. The necessary conditions for designing such fibers is given in this following equation (refer to the dispersion-compensating fibers tutorial)
Thus, the DCFs used for the compensation of dispersion slope in WDM systems also provide control of third-order dispersion for each channel. The only problem with DCFs is that their dispersion characteristics are not easily tunable. As a result, system performance can be easily compromised if link dispersion changes because of temperature or other environmental changes.
The tunable compensation of dispersion slope is possible through optical filters. Planar lightwave circuits based on cascaded MZ interferometric filters have proved quite successful because of the programmable nature of such filters. As early as 1996, such a filter was designed to have a dispersion slope of -15.8 ps/nm2 over a 170-GHz bandwidth and used to compensate third-order dispersion over 300 km of a dispersion-shifted fiber with β3 ≈ 0.05 ps/(km-nm2) at the operating wavelength. The figure below compares the pulse shapes at the fiber output observed with and without β3 compensation when a 2.1-ps pulse was transmitted over 100 km of such a fiber. The equalizing filter eliminates the oscillatory tail and reduces the width of the main peak from 3.4 to 2.8 ps. The residual increase in the pulse width from its input value of 2.1 ps is partly due to PMD.
Chirped fiber gratings are often preferred in practice because of their all-fiber nature. Long fiber gratings (~1 m) were developed by 1997 for this purpose. In 1998, a nonlinearly chirped fiber grating was capable of compensating the TOD over 6 nm for distances as long as 60 km. The cascading of several chirped gratings can provide a dispersion compensator that has arbitrary dispersion characteristics and is capable for compensating dispersion to all higher orders. Figure (a) below shows a simple configuration for compensating the TOD β3 of a fiber. Two identical chirped fiber gratings are cascaded through an optical circulator, but one of them is flipped over so that their chirps are opposite in nature. As the group-delay slopes are equal but of opposite signs for the two gratings, the combination provides no net GVD. However, their TOD contributions add up to produce a nearly parabolic shape for the relative group delay, as shown in figure (b) below.
An arrayed-waveguide grating or a sampled fiber grating can also compensate for second- and third-order dispersions simultaneously. Although a nonlinearly chirped sampled grating can provide tunable dispersion for several channels simultaneously, its bandwidth is still limited. An arrayed-waveguide grating in combination with a spatial phase filter can provide dispersion-slope compensation over a bandwidth as large as 8 THz and should be suitable for 40-Gb/s multichannel systems. The feasibility of transmitting a 100-Gb/s signal over 10,000 km has also been investigated using midspan optical phase conjugation in combination with third-order dispersion compensation.
Tunable compensation of dispersion slope can also be realized by integrating a segmented thin-film heater with a chirped fiber grating. In a 2004 experiment, a 4-cm-long grating was heated in a distributed fashion using 32 thin-film segments. A DCF module was used after the grating to ensure that accumulated second-order dispersion was zero at the center wavelength of the channel. It was possible to vary the dispersion slope from -20 to +20 ps/nm2 by adjusting temperature distribution along the grating. In a different approach, two fiber gratings, linearly or nonlinearly chirped by applying a strain, were cascaded in series through an optical circulator. Both grating were mounted on a substrate that could be bent by moving a block. It was possible to change only the dispersion slope from nearly 0 to -58 ps/nm2 over a bandwidth of 1.7 nm without affecting the Bragg wavelength of the grating.
Even unchirped fiber gratings can be used to realize tunable values of third-order dispersion. The figure below shows one design in which two unchirped gratings are mounted on a metal beam that can be bent by applying strain. A four-port circulator was used to send the input signal through each grating in a cascaded fashion. The third-order dispersion of such a device could be tuned by adjusting the applied nonlinear strain, without changing the second-order dispersion. In this 2009 experiment, dispersion slope could be varied from -13.9 to -54.8 ps/nm2 over a bandwidth of more than 2 nm.
Several experiments have explored the possibility of transmitting a single channel at bit rates of more than 200 Gb/s. In a 1996 experiment, a 400-Gb/s signal was transmitted by transmitting 0.98-ps pulses inside a 2.5-ps time slot. Without compensation of third-order dispersion, the pulse broadened to 2.3 ps after 40 km and exhibited a long oscillatory tail extending over 6 ps, a characteristic feature of the third-order dispersion. With partial compensation of third-order dispersion, the oscillatory tail disappeared, and the pulse width reduced to 1.6 ps, making it possible to recover the 400-Gb/s data with high accuracy. Optical pulses shorter than 0.5 ps were used in 1998 to realize a bit rate of 640 Gb/s. In a 2001 experiment, the bit rate was extended to 1.28 Tb/s by transmitting 380-fs pulses over 70 km of fiber. The propagation of such short pules requires compensation of second-, third-, and fourth-order dispersions simultaneously. The highest single-channel bit rate of 2.56 Tb/s was realized in a 2006 experiment in which a DQPSK signal was transmitted over 160 km.
3. PMD Compensation
As discussed in this dispersion in single-mode fibers tutorial, polarization mode dispersion (PMD) leads to distortion of optical pulses because of random variations in the birefringence of an optical fiber along its length. This distortion occurs in addition to GVD-induced pulse broadening. The use of dispersion management can eliminate GVD-induced broadening but does not affect the PMD-induced degradation of an optical signal. For this reason, the control of PMD has become a major issue for modern dispersion-managed lightwave systems.
Before discussing the PMD-compensation technique, it is important to obtain an order-of-magnitude estimate of the maximum link length for uncompensated systems. The root-mean-square (RMS) value of the differential group delay (DGD) for a link of length L is given by
where Dp is the PMD parameter. It is important to note that instantaneous values of DGD fluctuates with time over a wide range because of temperature and other environmental factors. If DGD becomes so large that it exceeds the bit slot, a lightwave system stops functioning properly; this is referred to as fading or outage in analogy with a similar effect occurring in radio systems.
The performance of a PMD-limited system is quantified using the concept of the outage probability, which should be below a prescribed value (often set near 10-5 or 5 min/year) for acceptable system performance.
An accurate estimate of outage probability requires extensive numerical simulations. In general, outage probability depends on the modulation format, among other thins. The following figure shows outage probability as a function of average DGD for the NRZ and RZ formats assuming that outage occurs when the power penalty exceeds 2 dB to maintain a BER of 10-12.
In general, the performance is better for RZ format with shorter pulses. The main conclusion is that the RMS value of DGD should only be a small fraction of the bit slot at a given bit rate B. The exact value of this fraction varies in the range of 0.1 to 0.15 depending on the modulation format and other design details of a lightwave system.
If we use 10% as a conservative criterion for this ratio and use BσT = 0.1, the system length and the bit rate are related to the PMD parameter Dp of the fiber by the condition
We can use this condition to estimate the maximum PMD-limited distance over which a system can operate at a given bit rate B. In the case of "old" fiber links installed using standard fibers, this condition becomes B2L < 104 (Gb/s)2-km, if we use 
as a representative value. Such fibers require PMD compensation at B = 10 Gb/s when the link length exceeds even 100 km. In contrast, modern fibers have typical Dp below 
. For systems designed using such fibers, B2L can exceed 106 (Gb/s)2-km. As a result, PMD compensation is not necessary at 10 Gb/s but may be required at 40 Gb/s if the link length exceeds 600 km. It should be stressed that the above expression provides only an order-of-magnitude estimate. Moreover, this condition can be relaxed when the technique of forward error correction (FEC) is employed at the receiver.
The preceding discussion shows that PMD can limit the performance of long-haul systems when the channel bit rate exceeds 10 Gb/s. For this reason, techniques for compensating PMD attracted attention as early as 1994 and have continued to evolve since then. Here, we focus on optical techniques; electrical techniques are covered in the next tutorial. The following figure shows the basic idea behind optical PMD compensation. It consists of a polarization controller followed with a birefringent element such as a polarization-maintaining fiber. A feedback loop that measures degree of polarization uses this information to adjust the polarization controller.
The performance of the simple PMD compensator shown in figure (a) above is limited by the fixed DGD provided by its birefringent element. Several other designs employ a variable DGD using a tunable delay line. The PMD-distorted signal is separated into its two components using a polarization controller and a polarization beam splitter. The two components are combined after introducing an adjustable delay in one branch through a variable delay line. A feedback loop is still needed to obtain an error signal that is used to adjust the polarization controller in response to environmental changes. The success of this technique depends on the ratio L/LPMD for a fiber of length L, where LPMD = (T0/Dp)2 and T0 is the pulse with. Considerable improvement is expected as long as this ratio does not exceed 4. Because LPMD is close to 10,000 km for 
and T0 = 10 ps, such a PMD compensator can work over transoceanic distances for 10-Gb/s systems.
PMD compensation can also be realized using devices that do not employ optical fibers. Examples include LiNbO3-based distributed compensator, ferroelectric liquid crystals, optical all-pass filters, birefringent chirped fiber gratings, and yttrium orthovanadate (YVO4) crystals. Figure (a) below shows a PMD compensator based on YVO4 crystals that has been used successfully for lightwave systems operating at a bit rate of 160 Gb/s. It consists of multiple birefringent YVO4 crystals of different lengths separated from each other by Faraday rotators. More precisely, the length of each crystal is one half of the previous one. Because of this feature and the use of tunable Faraday rotators, such a device was capable of providing tunable DGD ranging from 0.31 to 4.70 ps in steps of 0.63 ps. In the 160-Gb/s experiment, a LiNbO3-based polarization controller was used to scramble the state of polarization at the transmitter end and to ensure the detection of changes in instantaneous DGD at the receiver end.
Several other kinds of PMD compensators have been developed. A virtually imaged phase array, a device also sometimes used for GVD compensation, can be used for PMD compensation after suitable changes. Such a device converts the frequency-dependent Jones matrix of the fiber link that is responsible for PMD to a frequency-independent constant matrix in a dynamic fashion. In a recent 160-Gb/s experiment, a polarizer-based PMD compensator shown in figure (b) above was employed with success. In this device, the orthogonally polarized components of the optical signal are detected using photodiodes, and the resulting currents are used to adjust a polarization controller in a dynamic fashion.
It should be stressed that most PMD compensators help to mitigate only the first-order PMD effects. At high bit rates, optical pulses are short enough, and their spectrum becomes wide enough, that the PSPs cannot be assumed to remain constant over the whole pulse spectrum. Higher-order PMD effects become of concern for lightwave systems operating at bit rates of 40 Gb/s or more. The compensation of second- and even third-order PMD may be necessary in some cases. In most cases, a first-order PMD compensator can increase the tolerable value of DGD by more than a factor of 3, resulting in a substantial increase in the transmission distance for PMD-compensated systems. In practice, a separate PMD compensator is required for each channel. This fact makes PMD compensation along the fiber link a costly proposition for WDM systems. An electrical equalizer built into the receiver provide an alternative practical solution for both GVD and PMD compensation. We turn to this topic in the next tutorial.