9.2.2.1 Impact of Chromatic Dispersion on 100 Gbit/s Transmission

Chromatic dispersion is so a bad problem even at 10 Gbit/s that uncompensated systems are dispersion limited at about 36 km. However, due to the deterministic nature of this phenomenon, a carefully designed compensation map can almost completely cancel the problem at 10 Gbit/s, but for the limitation posed to the span length from the interplay between dispersion and nonlinear effects.

At 100 Gbit/s the situation is completely different. As a matter of fact, not only the second order dispersion has a big impact on transmission, but also the third order dispersion (the so-called dispersion slope) cannot be neglected.

The effect of the dispersion slope is evident from Figures 9.1 and 9.2. The result of the simulation of a perfectly linear transmission of a WDM 100 Gbit/s NRZ signal through a link of DSF fiber is reported in Figure 9.1 without polarization mode dispersion (PMD). The WDM comb is composed of 16 channels at 200 GHz spacing and the dispersion zero is exactly in correspondence with the carrier of the eighth channel.

In particular, the pulse shape of the 16th channel is reported for a dispersion slope of 0.05ps³/km and different link length. All the pulses are normalized to their maximum amplitude.

The pulse distortion due to the dispersion slope is quite evident, and after 5 km propagation, the greatest part of the transmitted energy is out of the bit interval while the pulse shape is completely destroyed.

This means that dispersion compensation has to be performed channel by channel, taking into account the exact dispersion value in the channel spectral position.

But this is not enough, as shown in Figure 9.2. In this figure, similar to Figure 9.1 for its construction, the center channel of the WDM comb is considered, having the dispersion zero in its center wavelength.

Nevertheless, the fact that the channel spectrum is so wide causes a pulse distortion due to the third order dispersion even if the average dispersion in the channel bandwidth is zero. The intersymbol interference (ISI) caused by third order dispersion is already important after 50 km propagation and after 200 km propagation a relevant quantity of the pulse energy is out of the bit interval.

The need of per channel compensation depends from the width of the NRZ spectrum that is about 200 GHz. Since multilevel modulation shrinks the spectrum of the transmitted signal, it helps to attenuate the penalty dependence on third order dispersion.

As an alternative, or even to integrate the effect of spectrum width reduction, per channel compensation can be applied.

The cheaper method to operate per channel dispersion compensation is to use an electronic equalization circuit. The exact compensation of a precise wavelength behavior of the dispersion coefficient is more difficult if electronic dispersion compensation is used in conjunction with Direct Detection since the nonlinear detector characteristic has to be taken into account. This difficulty is removed if coherent detection is used, due to the complete proportionality of the photocurrent to the optical field, both in amplitude and phase. In this condition, if linear fiber propagation can be assumed, the propagation channel is linear even considering the detector characteristic, thus allowing far more efficient dispersion compensation to be applied.

9.2.2.2 Tunable Optical Dispersion Compensator

A great amount of research has been devoted also to Tunable Optical Dispersion Compensators (TODC) for application at 40 and 100 Gbit/s. A TODC is an optical transparent component that is able to compensate a selectable amount of dispersion (generally with a selected slope) on one or more high speed channels.

TODC are divided into single channel and multiple channels devices, where the first are suitable to compensate a single channel, while a whole WDM comb can be simultaneously compensated by the latter.

Single channel TODC [1] are mainly studied for intermediate reach, very high speed systems, where electronic dispersion compensation is not enough due to the nonlinear characteristics of the photodiode, but cost constraints or other considerations (related for example to real estate or power consumption) render unsuitable coherent detection.

On the other hand, the advantage expected by multichannel TODC is the ability to compensate the dispersion of a whole WDM comb with a single optical device, without the need of demultiplexing the optical channels.

Thus, multichannel TODCs are complementary to the electronic compensation and the most probable solution to deal with chromatic dispersion of large bandwidth optical signals is a contemporary use of TODC for in-line compensation plus electronic fine equalization at the receiver.

This solution, based on periodic compensation along the optical transmission line, has the advantage of avoiding combination between dispersion and nonlinear distortion that unavoidably occurs if all equalization is placed at the receiver and that renders electronic equalization much more difficult and expensive.

In order to be used in this way, the multichannel TODC have to be simple from a design point of view and must be able to compensate the typical amount of dispersion occurring in a long haul system span.

One of the more direct ways to realize a multiple channel TODC is shown in Figure 9.3 and consists in dividing the incoming optical channel between the two branches of an interferometer where two different dispersion elements are placed. The amount of optical power sent in each branch is regulated by a selectable directional coupler [2].

The transfer function of the system is given by

(9.2)

where

k_i indicates the coefficient of dispersion of the dispersive elements

τ_i is the overall delay along the two interferometer branches

α is the tunable splitting ratio, while the factor 1/2 takes into account the unavoidable loss of the beam splitter

The module of the transfer function writes

(9.3)

where Δk = k₁ − k₂ and Δτ = τ ₁ − τ₂.

The module of the spectral response of the interferometer is a periodic function, as obvious, whose period is frequency dependent due to the quadratic behavior of the object of the cosine function. The optical bandwidth depends on the frequency dependent term and it is maximum where this term is zero, that is, when α = 0 or α = 1.

On the other hand, the optical bandwidth is minimized when the third term in Equation 9.3 is maximum, that is, α=1/2.

In order to determine the group delay of the interferometer and then the overall dispersion characteristic, the phase contribution Φ(ω) of the response in Equation 9.2 has to be calculated and derived with respect to the angular frequency.

The result depends again on the factor A(ω) = (Δkω² + Δτω) that appears under a cosine function.

It is possible to select Δk and Δτ so that in a determined optical bandwidth, the approximation is verified: A(ω) = (Δkω² + Δτω) = 0.

Assuming to be in this condition, the differential delay expression of the interferometer can be quite simplified, up to arriving at the following expression

(9.4)

Equation 9.4 has the form expected by a TODC, since it linear in ω and depends on an adjustable parameter, which is α.

Just to make an example, let us consider a fiber interferometer where the dispersion is achieved by fiber brag gratings.

The device parameters are summarized in Table 9.1. From the table it results that the condition of A(ω) = 0 is attained introducing a length difference of about 1.1 μm among the gratings at a wavelength of 1.55 μm. The behavior of cos[A(ω)] is represented in Figure 9.4 versus the wavelength deviation with respect to 1.55 μm.

TABLE 9.1 Physical Parameters of the Fiber Components Used in the Example of Wideband Tunable Dispersion Compensator

The TODC in this example has a bandwidth of about 40 nm around the reference wavelength (corresponding to 1.55 μm) that is a very good bandwidth to operate as multi-wavelength TODC in DWDM systems.

The group delay is shown in Figure 9.5 versus the wavelength deviation for different values of α. The group delay behavior is, in this ideal case, linear with the wavelength with a very good approximation, thus confirming the good performance of this architecture in conjunction with fiber Brag gratings.

Naturally, in practical implementations, several nonidealities affect the behavior of the TODC, like reflectivity of the gratings, ripple in the phase characteristics of the dispersion elements, losses of the whole device, and so on. A detailed analysis of such impairments besides a practical implementation of a compensator and a transmission experiment are presented in [2].

Moreover in [2] it is also shown that a compensator with the architecture of Figure 9.3 can also be used as a single channel compensator coupling the signal directly in the electrical domain. In this case, the compensator is an alternative to electronic compensation.

Several other architectures have been proposed for the TODC, having different characteristics.

A class of compact TODC are based on the use of Array Waveguides (AWGs) [3,4] since these devices provide contemporary demultiplexing and different dispersion contribution channel per channel. Moreover, if suitably designed, the same AWG can be used both for multiplexing and for demultiplexing the DWDM comb increasing the dispersion contribution and gaining in compactness.

An example of AWG architecture is presented in [3] and is reported in Figure 9.6. The component working is quite simple. The incoming optical field is decomposed by the AWG, which divides its spectrum slices, its spectral phase around the center frequency is adjusted by the lens located near the spectral plane, reflected by the mirror, and recombined in the AWG to regenerate the field. Depending on the AWG design, this TODC can be used either to compensate interchannel dispersion due to the dispersion slope in particularly wide spectrum transmissions or multichannel dispersion in WDM systems.

In the first case however, the distortion due to the spectral analysis of the channel and successive composition of the spectral components has to be taken into account. The compensator tuning is performed by using a cooler and exploiting the thermal dependence of the optical characteristics of the silica composing the AWG and the lens structure.

An improvement of the performance of the TODC can be attained by realizing several lenses monolithically integrated into the structure of the chip so that the whole device is a planar silicon on silica-integrated circuit. The characteristics of an experimental realization of the device are reported in Figure 9.7 [3].

Another technology used to implement experimental TODCs is based on discrete optics and on technologies similar that used in 3D Micro Electrical Mechanical machines (MEMs), Liquid Crystal on Silicon (LCOS) switches and in Wavelength Selective Switch (WSS) components (see Chapter 5 for MEMs switches and Chapter 6 for WSS).

An example of TODC using this technology is reported in [5] and its block scheme is shown in Figure 9.8. A circulator deviates the whole DWDM comb into an AWG where the channels are demultiplexed. At the AWG output, a free space optics directs the demultiplexed channels on a bulk grating and, at the grating output, a lens focalizes the channels on an LCOS switch. The LCOS reflects back the channels that traverse the system again in the opposite direction to be reinjected by the circulator into the transmitting fiber after multiplexing by the AWG.

The AWG, the LCOS deflector, and the Brag grating, all provide dispersion and, if the dispersion axis of the grating is orthogonal to that of the AWG and the two dispersion axes of the LCOS are parallel and orthogonal to the AWG axis, it is possible to combine the dispersion contributions to have different dispersions for the different channels.

In the paper where this architecture is proposed [5] the TODC design is performed for application in the L band at 40 Gbit/s.

The measured dispersion performances for a single channel among the 50 that can be simultaneously processed by the device are reported in Figure 9.9 for different driver voltages of the LCOS. The ripple that can be observed in the experimental characteristics is attributed to the imperfect alignment of the LCOS device, a hypothesis that is reinforced by the presence of an analogous ripple in the amplitude transmittance [5]; in any case, the ability of the device to attain a range of dispersion between −800 and 800 ps/nm results clearly from the figure.

9.2.3.1 Impact of Polarization Mode Dispersion on 100 Gbit/s Transmission

A first evaluation of the impact of PMD on transmission of high speed channels has been carried out in Chapter 6 and an example of performance evaluation is reported in Figure 6.12 for channels at 40 and at 100 Gbit/s.

Using the parameters of Figure 6.12 (see Table 6.1) it is possible to evaluate the transmission length at which the PMD-induced penalty is 2 dB, that we can call maximum PMD transmission distance. The value of the maximum PMD transmission distance is reported in Figure 9.10 versus the bit rate for different values of the PMD parameter. Dots are values calculated with the model of Chapter 6 and assuming linear propagation. The curves are interpolated with a function inversely proportional to the bit rate through a constant.

From the figure it is quite evident that the fit is very good, almost in all the interesting range of values for the PMD parameter.

Thus we can conclude without a big error that the maximum PMD transmission distance is inversely proportional to the bit rate.

From Figure 9.10 it is also clear that the impact of PMD is very important at 100 Gbit/s: the maximum PMD transmission distance for a PMD parameter of 0.2 ps/km^1/2 is about 100 km.

Also in this case, since the dependence of the PMD penalty on the bit rate is due to the relationship between the bit rate and the optical bandwidth, both adoption of multilevel modulation and coherent detection are beneficial for the same reasons discussed in the above section.

9.2.3.2 Polarization Mode Dispersion Compensation

PMD can be naturally compensated both electronically and optically. The main issue in PMD compensation is the fact that it is a random phenomenon, thus requiring adaptive compensators to follow its fluctuations.

Electronic compensation circuits generally have such a feature, thus being suitable for compensating PMD, besides other ISI-related impairments like chromatic dispersion. For example, in Figure 5.53, it is shown as both feedforward equalizer/decision feedback equalizer (FFE/DFE) and maximum-likelihood estimation (MLE)-based compensators can reduce the PMD-induced penalty for 10 and 20 Gbit/s signals.

Scaling the bit rate at 100 Gbit/s poses however a great challenge to electronic compensation, due to the great speed at which the compensator has to work.

Even if traveling wave FFE/DFE compensators have been realized and demonstrated as effective in compensating PMD at 40 Gbit/s [6], no electronic equalization has been realized on a serial signal at 100 Gbit/s.

For this reason, optical PMD compensation has a particular importance at very high speed.

As a matter of fact, due to the random nature of the PMD, the instantaneous performance of the system can be also completely ruined by PMD in the occasion of unfortunate fluctuations of the Jones matrix even if the average penalty is quite low.

To understand this phenomenon, let us consider the PMD-related penalty as defined in Figure 6.12: it was evaluated by starting from the error probability for a specific value of the PMD broadening Δτ and averaging with respect to Δτ. It is clear that, while Δτ fluctuates, the system can experience instantaneous values of the penalty either smaller or larger than the average value.

To evaluate the impact of these fluctuations, generally, the so-called outage probability P_O is used, that is, the probability that the instantaneous value of the PMD-induced penalty is higher than the value used in the system design, or equivalently (assuming all processes ergodic), the percentage of time in which the PMD-induced penalty is higher than the design value.

In order to approach the problem of reducing the outage probability, it is important to notice that when launching a pulse into a fiber link with PMD, the received pulse width depends on the launched state of polarization (SOP), and in particular on its position with respect to the input Principal States of Polarization (PSPs), following Equation 4.29. The effect of the launched polarization state on the pulse broadening is represented in Figure 9.11 [7] where the pulse spreading ratio β is shown versus the normalized average pulse width α.

The parameters β and α are defined as

(9.5)

where

τ_in is the half width at half maximum of the input pulse

〈τ〉 is the average half width at half maximum of the output pulse

σΔτ2 is the variance of the half width at half maximum of the output pulse as defined in Equation 4.29

Looking at Figure 9.11, it results that there is quite a difference between launching the input pulse along the worst and along the best polarization state. Thus, a way to attenuate the penalty due to PMD via the elimination of the worst cases is simply to apply a fast random polarization scrambling.

If the polarization scans almost all the possible states during a bit period, the pulse spreading will be, in a very first approximation, always the average one and the worst cases will be avoided.

In order to evaluate the effectiveness of this method, let us image that the outage probability is P_O = 10⁻³ in the absence of fast polarization scrambling and that, in order to avoid mixing of the PMD effect with self-phase modulation (SPM), we will introduce periodic scrambling along a 300 km long line at 100 Gbit/s with a PMD of 0.1 ps/km^1/2 (see Figure 9.11).

The effect on the outage probability of periodic polarization scrambling via equally spaced scramblers placed along the line is shown in Figure 9.12 versus the number of scramblers [8].

Polarization scrambling technique is quite effective in reducing penalty fluctuations due to the random nature of the PMD, with the additional advantage of being able to process a whole WDM comb without demodulation, but while the bit rate increases, it is more and more difficult to perform a perfect scrambling, that is, scramble at a so high speed that almost all the polarizations states are traversed during the bit time. Let us imagine that the scrambling speed is not sufficient, in particular, let us assume that it is 20% smaller than the speed needed to span all the possible states of polarization.

The resulting outage probability is also reported in Figure 9.13 [8] and the result is that, even if the scrambling is much less effective, the outage probability is in any case much smaller than the case without scrambling if a sufficient number of scramblers is used.

The fact that PMD fluctuations induce outage also suggests the fact that errors due to these fluctuations form bursts. In case of transmission channels causing error bursts, the use of specific burst correcting forward error correcting codes (FECs) is quite effective.

Systems using specific FECs and fast polarization scrambling can be optimized to greatly reduce PMD penalty fluctuations up to very low values of the outage probability [9,10].

If the average penalty due to PMD is acceptable from a system design point of view, a reduction of the outage probability below a required lower limit is enough to assure a correct system working.

If the average PMD penalty is too high, an effective PMD compensation is needed that is able to reduce both the outage probability and the average penalty. The scheme of a simple PMD compensator, a so-called first order compensator, is reported in Figure 9.13. At the transmission fiber output a Polarization Controller is set to split the signal between fast (p→−) and slow (p→+) PSP components. The components along the PSPs are mutually delayed by the differential group delay (DGD) Δτ and then recombined. The delay is generally obtained by rotating the PSPs to match the axes of a tunable birefringence element.

Both the PSP decomposition and the DGD have to be adjusted via an adaptive algorithm to match the slowly varying transmission link characteristic [11–14].

Several algorithms have been devised to simplify the apparent complexity of the PMD compensator and several designs have been presented that are quite effective in perfectly compensating the PMD in the central frequency of the signal spectrum.

Obviously, to compensate a whole WDM comb, it has to be demultiplexed and compensated channel by channel. Thus, a first order PMD compensator like that shown in Figure 9.13 is generally designed to be deployed immediately before detection.

A first order PMD compensator has two main limitations. The first is related to the nonlinear evolution of the signal along very long fiber links. When the linear and nonlinear polarization evolutions are mixed, it is not yet possible to distinguish them and a linear compensator is not yet able to correctly recover the signal [11]. This means that in order to use effectively such compensators, the propagation has to be maintained as linear as possible (but for the soliton case that is considered in detail in [12]).

More important, by increasing the bit rate, the bandwidth of the PSPs starts to be of the order or even smaller than the optical bandwidth of the signal. As a matter of fact, from Chapter 4, we derive that the PSP bandwidth in a Standard Single Mode Fiber (SSMF) if of the order of 100 GHz, while the bilateral optical bandwidth of a 40 Gbit/s NRZ signal is 80 GHz and that of a 100 Gbit/s NRZ signal is 200 GHz.

In such a condition, intrachannel variations of the PSP causes a relevant signal distortion even if a first order PMD compensator is used.

A more accurate compensation can be attained by approximating the PMD with a second order expression in ω instead of a first order one [15] and designing a second order PMD compensator. A possible scheme of a second order PMD compensator is reported in Figure 9.14 [16,17].

The effectiveness of second order compensation is quite improved for high bit rate signals with respect to first order compensation as shown in [18]. A comparison between different first order and second order PMD compensators based on various control algorithms and compensating elements is reported in Figure 9.15 [18] where the outage probability P_O is reported as a function of the design average PMD penalty ΔSN_e (dB) for a bit rate of 100 Gbit/s, NRZ transmission, and an average PMD-induced DGD of 3.8 ps (deriving, e.g., from a 100 km long link with D_PMD = 0.38 ps/km^1/2).

Comparing Figure 9.15 with literature, it is to take into account that the penalty is frequently expressed as instantaneous Eye Opening Penalty (EOP). Since evolution of the PMD is much slower than the bit-time, the PMD DGD can be considered almost constant for a great number of consecutive bits. In this situation, there is a direct relationship between the instantaneous EOP and the SN_e penalty once the link characteristics are fixed. As a matter of fact, indicating with I₀ and I₁ the average values of the photocurrent decision samples corresponding to the two transmitted bits, and with I₀₀ and I₁₀, the same values in the absence of PMD, we can write (see Chapter 6)

(9.6)

where σ₁ and σ₀ are the overall noise variance corresponding to the two transmitted bits.

Assuming that the noise does not change, the following equation is immediately derived:

(9.7)

The PMD compensators considered in Figure 9.15 are

• Conventional first order-polarization mode dispersion compensation (FO-PMDC) with variable differential phase delay

• PMDC based on the Kogelnik–Bruyère model described in [19]

• PMDC based on the truncated exponential expansion form (EEF) described in [20]

• PMDC described by [18]

• PMDC formed by two concatenated First Order-PMDCs, wherein the first introduces a fixed differential phase detector (DPD) equal to the mean DGD in the fiber and the second a continuously variable DPD [21]

• PMDC described by [19,21]

The performance of second order compensators depends in an important measure on the control algorithm and on the effectiveness of the compensation device, but it is in any case quite better than the performance of a single order compensator.

The number of stages of the PMD compensator can be increased to three or more in order to achieve high order compensation and target better performances. More compensation stages however also mean a more complex control algorithm and a more expensive device (keeping in mind that PMDCs are per channel devices).

The performance improvement and the control complexity of high order compensation devices have been studied in several papers [22]. An estimation of the possible performance of a third order PMD compensator is reported in Figure 9.16 [22]. When compared to a relevant gain passing from a first to a second order compensator, the gain achieved when passing from a second to a third order component is not so pronounced while it is paid with a sensible increase in the complexity. Thus it is possible to conclude that system design should try as much as possible to use second order compensator (if optical compensation is needed) and passing to third order compensators only if it is really needed.

9.2.4.1 Fiber Nonlinear Propagation

Increasing the signal bandwidth has essentially no impact both on Brillouin (at least if the carrier is maintained in the spectrum) and on Raman effects.

While the impact of Cross Phase Modulation (XPM) and Four Wave Mixing (FWM) can be managed by a suitable channel spacing and residual dispersion, SPM depends on the individual channel and can become the dominant nonlinearity and the system limiting factor besides PMD.

In this case also, the reduction of the signal bandwidth occupancy seems to be the only way to go to reduce SPM impact besides the possibility of an effective electronic compensation via some kind of coherent detection or of signal predistortion (compare Chapter 5).

9.2.4.2 Timing Jitter

Due to the very short bit time, timing jitter assumes a particular importance in 100 Gbit/s serial systems. We can decompose the jitter into three factors:

1. Transmission jitter

2. Detection clock jitter

3. Propagation jitter

The transmission jitter depends on the fluctuations of the transmitter clock that causes the center instant of the pulses sent to the modulator to fluctuate randomly around the average value. This problem can be faced with a combination of a local improvement of the clock electronics and a more performing network timing distribution in case synchronous timedivision multiplexing (TDM) is used.

Analogously, the receiver clock jitter depends on the imperfection of the receiver clock electronics that causes the sampling instant before the decision device to fluctuate randomly around the average value. Also in this case, the problem can be faced with a combination of a local improvement of the clock electronics and a more performing network timing distribution.

The propagation jitter is caused by a different phenomenon and affects systems where SPM is partially compensated by a residue chromatics dispersion [23]. This jitter contribution becomes evident for pulses with a full width at half maximum (FWHM) of the order of few ns, that could be used in return-to-zero (RZ) 100 Gbit/s systems.

The propagation jitter is a combination of the jitter due to pulse interaction and to the jitter due to ASE to pulse nonlinear combination.

The first effect is a reduced form of the Gordon-Haus jitter that appears in soliton propagation. This jitter caused by attraction between nearby pulses becomes more and more effective by increasing the Chromatic Dispersion–SPM compensation degree [24,25], and comes into play especially when strong under-compensation of chromatic dispersion is used to counterbalance SPM. The other cause of propagation jitter is the phase noise produced by the nonlinear ASE to signal coupling and it is known as Gordon-Mollenauer effect [26].

9.2.4.3 Electrical Front End Adaptation

A serial 100 Gbit/s signal has a huge bandwidth and it is not easy to device receivers with a low-pass bandwidth of 100 GHz. Distortions in the photocurrent directly affect the receiver performances via ISI and loss of power in the bit interval. Great progress have been made in the last years, but this is yet an important challenge [27].

9.3.2.1 Multilevel Differential Phase Modulation (M-DPSK)

This is probably one of the simpler cases that generalize to multilevel transmission binary differential phase shift keying (DPSK) that we have introduced in Chapter 6: the phase of the transmitted signal is modulated using multilevel differential modulation. Let us call w_j the jth world of n transmitted bits and Δθ = 2π/2ⁿ .

The M-DPSK signal transmitted in the kth symbol interval and modulated with w_j writes

(9.16)

where η_T(t,k) is the Heaviside function that is equal to one for t ε (k − 1)T, kT) and zero elsewhere and θ(k) − θ(k − 1) = jΔθ.

As in the binary case, the name differential comes from the fact that the transmitted symbol is encoded into the phase difference between signals in two consecutive symbol intervals.

In terms of quadrature components, (9.16) writes

(9.17)

Equation 9.17 shows that the points representing the transmitted symbols in the quadrature space are all distributed on a circle on the e₃, e₄ plane whose center is in the origin and whose square radius is the transmitted power, which is constant due to the pure phase modulation.

A typical characteristic of a multilevel constellation is the minimum distance d in the quadrature space between a couple of symbols. It is quite intuitive that this distance will be related to the noise robustness of the modulation format. Considering for example binary IM, the constellation is done by two points and the distance is d=P, so that we can write SN_o = d ²/P_ASE, where P_ASE is the ASE power.

In this case, we have simply

(9.18)

A practical transmitter for the M-DPSK signal is presented in Figure 9.18, conceptually, this is simply a phase modulator, in practice almost only dual drive integrated modulators are used to reduce distortions (see Chapter 5).

9.3.2.2 Multilevel Quadrature Amplitude Modulation (M-QAM)

This modulation format is widely used in radio applications and consists in the amplitude modulation of the two complex quadratures of one of the field polarizations.

The points representing the states of the transmitted field in the quadratures space are comprised in one coordinate plane and form a square lattice: a few examples are provided in Figure 9.19.

In terms of quadrature, the components of the transmitted field can be written as

(9.19)

where s_j,k is the amplitude of the symbol transmitted along the jth quadrature in the kth symbol interval.

The maximum distance of QAM modulation depends on the adopted pattern and it can be easily evaluated starting from the fact that the power of an optical field is the square of its distance from origin. The minimum distance for a few common QAM patterns is reported in Table 9.2.

A general scheme for an optical M-QAM modulator is reported in Figure 9.20: a laser emits a linearly polarized optical field that is divided into two fields with a phase difference equal to π by a directional coupler. Each field is independently modulated by an amplitude modulator and then one of them is delayed to introduce a further phase difference of π/2. At that point, the two fields are combined again by a directional coupler that introduces a further phase difference of π, so that the total phase difference if 2π + π/2 and the two combined fields are exactly the quadratures of the M-QAM signal.

Such a scheme is suitable to be integrated in a single optical chip, for example, using lithium-niobate or indium phosphide (InP) technology.

Several different modulators have been proposed for particular M-QAM constellations. An interesting scheme is that of the hybrid modulator for 64-QAM proposed in [32]. The principal scheme of the modulator is shown in Figure 9.21 besides the way in which the 64-QAM constellation is formed in the different stages of the modulator. The fact that the modulator is composed of a parallel of three dual drivers phase modulators that has to be driven to produce a four-level phase modulated signal renders the modulator control easy and the loss relatively small.

TABLE 9.2 Normalized Constellation Minimum Distance for M-QAM

The most interesting property of this modulator is, however, that it can be realized by using hybrid integration with two Silica on Silicon chips and a modulator array composed of three phase modulators (let us remember that the modulator of Figure 9.21 is only apparently simpler, since each amplitude modulator is in reality a couple of phase modulators on the branches of an interferometer). The scheme of the hybrid integrated chip is reported in Figure 9.22 where the Silica on Silicon chips are in white and the Lithium–Niobate chip is in gray.

With the same technology, also other kinds of multilevel modulators has been designed and prototyped, like a 16-QAM modulator [33] and a 4-DPSK modulator [34] demonstrating its flexibility.

9.3.2.3 Multilevel Polarization Modulation (M-PolSK)

This modulation format exploits the property of the optical field to be polarized to switch the field polarization among a certain number of states that represents the transmitted symbols. In order to represent optical field states with the same power and phase, but with different polarization in the quadrature space it is needed to represent the surface defined by the following equations:

(9.20)

where the three angles φ and θ define the field polarization. Naturally, the surface is a part of a three-sphere due to the characteristic of the fields described by Equation 9.20 to have the same power. In particular, it is that part corresponding to all the points with the same absolute phase.

Due to the characteristics of this surface, it is not so easy to use quadrature representation to describe polarization modulation.

In optics however, when there is the need for describing the polarization evolution, the Stokes parameters representation is frequently used.

The Stokes parameters are defined as

(9.21)

It is easy to verify that, for a monochromatic perfectly polarized beam it is

(9.22)

so that the SOP of a field can be represented as a point in the so-called Stokes space, which is an Euclidean three dimensional space having the three Stokes parameters as coordinates. In this space, the equal power surface is again a sphere, as in the quadrature space, but the sphere radius in this case is equal to the square of the field power. The sphere of normalized radius (assumed to be equal to one) in the Stokes space is called the Poincare sphere.

The position on the Poincare sphere of the main SOPs are shown in Figure 9.23.

These positions can be derived easily by rewriting the Stokes parameters expressions using spherical coordinates in the Stokes space as follows

(9.23)

The angular coordinates in the Stokes space, if defined as in Equation 9.23 and in Figure 9.23 are related to the angles determining the form of the polarization ellipse as shown in Figure 9.24 as it is possible to demonstrate starting from the definition (9.23) of the Stokes parameters.

From (9.23) and the angles definition of Figure 9.24, the position on the Poincare sphere of any SOP can be individuated.

Up to now we have supposed the field completely polarized, as an M-PolSK signal at the output of the modulator.

During propagation however, ASE is added to the field. The noise is randomly distributed among the two polarizations, thus introducing a field depolarization. In this case, the polarization degree can be defined as

(9.24)

Where P is the overall field power and the definition of the Stokes parameters becomes

(9.25)

If the light is composed of a fraction of perfectly polarized light and a fraction of perfectly depolarized light (like a PolSK signal plus ASE noise), the power of the polarized component P_p and the power of the unpolarized component P_u are given by

(9.26)

(9.27)

so that, if the Optical SN coincides with the ratio between the polarized and the unpolarized part of the field, it can be written as

(9.28)

The general instantaneous multilevel Polarization modulation format (M-PolSK: Multilevel Polarization Shift Keying) is obtained by choosing a constellation on the Poincare sphere and associating to each of the constellation points a transmitted symbol.

We will introduce in the following a Stokes parameters receiver for the M-PolSK [35] that can be demonstrated to be the optimum receiver for this type of modulation format [36]. We will see that, also in this case, what is relevant to evaluate the effectiveness of the constellation is the minimum distance in the Stokes space.

Limiting our analysis to constant power modulation, the problem of determining the best possible constellation can be formulated as the problem of distributing M points on a sphere so that the minimum distance between two of them is maximized.

This problem is not easy to be solved analytically and, as far as I know, there is no closed form solution for a generic number of points. A numerical solution for M = 4, 8, and 16 is presented in [37] where both the angular coordinates indicating the position of the points on the Poincarè sphere and the normalized distance matrixes are reported.

The optimum constellations are in a certain sense regular, but do not reproduce the vertices of polyhedrons or other known tridimensional forms. On the other hand, in [36] regular constellations are proposed for M-PolSK.

Considering the vertices of regular polyhedron in the space, the possible M-PolSK formats the minimum distance in the Stokes space and the minimum distance for the optimum configuration with the same number of points are listed in Table 9.3.

To construct regular constellations, Pythagorean and Archimedean solids are used as appropriates to achieve a given number of symbols. When several such solids exist with the same number of vertices, the best from the minimum distance point of view is selected [38,39].

Two of the less known regular solids cited in Table 9.3 are represented both in plane development and in tridimensional representation in Figure 9.25.

For the only three cases in which the optimum constellation is approximately known, it is evident that the gain in terms of minimum distance is relatively small. This will bring us to consider from now on the regular configurations listed in Table 9.3.

For each regular constellation, the power spectral density of the transmitted signal along two reference orthogonal polarizations can be evaluated with standard methods and the complete expression is reported in [37].

The way in which the transmitted power is divided among the two polarization components depends obviously on the choice of the polarization reference frame and, if the field at the output of a transmission fiber is considered, on the fiber instantaneous Jones matrix.

However, the power spectral density in any case exhibits a central zone comprised between the first two spectrum zeros, which are always symmetric with respect to the origin, where almost 92% of the power is comprised. The two considered zeros are at ω = ± 2πR/M thus confirming also for M-PolSK the general rule that the unilateral optical bandwidth can be assumed equal to B_o = R/M so that the spectrum width is 2R/M.

Under the point of view of modulation feasibility, a simple polarization modulator is needed to produce a PolSK signal. A polarization modulator can be realized using electro-optics materials exactly the same way a phase modulator is realized (see Chapter 5) [36,40].

Polarization modulation can be also implemented in conjunction with differential encoding. In this case, the information is coded into the change of the polarization vector in the Stokes space between two adjacent symbol intervals [41,42].

This further coding does not alter the structure of the transmitter, but for the addition of the differential encoder, and in a few cases is quite useful to simplify the receiver.

9.3.2.4 Multilevel Four Quadrature Amplitude Modulation (M-4QAM)

This modulation format exploits the possibility of modulating all the optical field degrees of freedom. The generic M-4QAM signal can be written as

(9.29)

where s_j,k is the amplitude of the symbol transmitted along the jth quadrature in the kth symbol interval.

A set of constellations for M-4QAM can be obtained simply replying on the two coordinates couples corresponding to the two polarizations the reference constellations of M-QAM.

It is not difficult to demonstrate that this procedure does not produce the optimum constellations from a minimum distance point of view, but is useful to simplify the modulator, that in this case is really a key issue.

As a matter of fact, if these configurations are chosen, the M-4QAM signal can be obtained by a modulator with the block scheme reported in Figure 9.26, which is constituted by two parallel M–QAM modulators.

Since M-QAM modulators can be built as in Figure 9.22, the same technology can be used for M–4QAM modulators.

If such regular constellations are chosen, the normalized amplitudes s_j,k of the four quadratures are extracted from four sets X_j whose elements can be expressed as a function of four integers numbers L_j and four indices h_j, as follows:

(9.30)

It is clear that, from (9.30) it derives

(9.31)

and that the ratio between the minimum distance d and the transmitted power P of a M-4QAM constellation is given by

(9.32)

A few key parameters of some M-4QAM formats of this class are summarized in Table 9.4. For comparison, in the table data relative to M-QAM are also reported. It results that using M-4QAM is always advantageous, but the gain gets smaller and smaller while M increases if squared configurations are used. Just as an example, for M = 16 the gain of squared 16-4QAM versus 16-QAM is 3.2 dB while it is reduced to 0.7 dB if M = 256.

If it is important to gain as much as possible from the constellation points distribution a better constellation has to be selected.

A great amount of work has been done by mathematicians to study forms and properties of four dimensional constellations [43,44].

TABLE 9.4 Minimum Normalized Distance in the Quadrature Space for Squared M-4QAM Constellations

M-QAM distances are reported for comparison.

In the particular case of constant power modulation we can use to define the M-4QAM constellations the vertices of convex, regular polyhedrons in four dimensions, as defined in [45]. Table 9.5 reproduces the structure of Table 9.4, but using the new M-4QAM constellations.

Even better results can be achieved with a careful optimization of more complex constellations [46,47].

It is out of the scope of this book to discuss in detail all the implications of the optimum choice of a four dimensional constellation, here we simply limit ourselves to a few examples based on a simple principle: creating a dense lattice of points via the Cartesian product of bidimensional QAM constellations and then select a subset of points for the final four dimensional constellation.

We will use a simple rule to construct our constellation as the Cartesian product of partitioned QAM constellations in which points are alternatively taken and discarded. An example of this kind of constellation is reported in Figure 9.27, where the generic M-QAM constellation is divided into two partitions: gray points that are taken as components in the (e₁, e₂) plane and black points that are taken as components in the (e₃, e₄) plane.

The minimum distances relative to the constellations that are built with this algorithm are reported in Table 9.6.

The results reported in Tables 9.4 through 9.6, besides the outcome of Equation 9.18 for M-DPSK are compared in Figure 9.28. An interesting observation emerges from this figure, which is useful to guide the decision about the number of levels to select when designing a multilevel system based on quadrature modulation. Increasing the number of available degrees of freedom passing from M-QAM to constant power M-4QAM to full M-4QAM is effective in maintaining an high distance among constellation points at high number of levels, while if the number of levels is low, a constellation with a lesser number of degrees of freedom can perform sufficiently well.

9.3.3.1 Four Quadrature Receiver

The block scheme of the four quadrature receiver is reported in Figure 9.29. The scope of the receiver is to detect the transmitted quadrature vector for each symbol interval and compare it with the constellation in the symbol space to produce an estimate of the transmitted symbol.

The receiver is divided into two sections: an optical front end (depicted in Figure 9.29a), whose scope is to detect the received field quadratures, and an electronic processing stage (whose architecture is reported in Figure 9.29b), whose scope is to invert the effect of fiber propagation to reconstruct the transmitted field.

A key component in the optical front end is the π/2 optical hybrid that is used, after the separation of the incoming field linear polarization components, both to insulate the quadratures of each polarization and to combine the LO with the incoming field before balanced detection.

Optical hybrids has been produced with several technologies [48–50] and their performances are quite good and stable.

The four fields E→j(j=1,2,3,4) obtained after polarization splitting and branching with π/2 hybrids can be written as

(9.33)

where the information bearing quadratures are indicated with ξ_j(t) and the ASE optical noise quadratures with η_j(t). The linear polarization unitary polarization vectors directed along the quadrature space reference axes are indicated with ζ→j(j=1,2,3,4) while the ASE polarization vector, that is a fast function of time due to ASE complete depolarization, is indicated with κ→(t). Finally “.” indicates the scalar product.

The field emitted by the LO at π/4 with respect to the local receiver axis is divided into its polarization components by a polarization beam splitter and coupled with the received field components by a set of four π/2 optical hybrids.

In a practical receiver, the set of hybrids can be substituted by a single monolithic hybrid realized in silica on silicon technology. A possible architecture for such a hybrid is shown in Figure 9.30.

Also the set of eight PIN Photodiodes that are needed to realize four balanced receivers can be realized with a diode bar in a single package so that the cost is not eight times the cost of a PIN diode but, for high volumes, could be less than two times.

At the exit of the balanced receivers, the four photocurrents are modulated at the intermediate frequency Δω and further electronic demodulation allows the four field quadratures to be derived.

In order to allow this operation to be performed with high performances, the intermediate frequency Δω has to be stabilized. This can be achieved by a frequency lock loop driven by the wavelength of the incoming signal [31,51].

If the LO frequency is exactly locked to the frequency of the incoming signal the IF demodulation is not needed and the photocurrents are directly in baseband. In this case, the coherent receiver is called homodyne [31].

Under a processing point of view, heterodyne receiver is more complex, but it has been used in radio frequency systems for a long time to alleviate the difficulty of designing wideband components with a very low low-pass frequency (ideally working up to continuous wave [CW]). In optical receivers, heterodyne detection was proposed for the same reason when coherent receivers was first proposed.

With the progress of electronics and with the possibility of digital design also for analog functions, this difficulty is no more so important and the simple homodyne structure seems preferable.

For this reason, up to now we will assume Δω = 0. In this case, the electronics front end reduces to a baseband filter fixing the electrical bandwidth B_e ≈ R_S of the receiver, R_S being the symbol rate, not differently from the case of a direct detection receiver.

The photocurrents in the homodyne receiver have the following expressions:

(9.34)

where

R is the photodiodes responsivity, which is assumed to be the same for all the photodiodes

P_L is the LO power, which can be written as P_L (t) = P_L0 [1 + r (t)] if the LO RIN r (t) has to be considered (see Chapter 5)

n_j(t) is the shot noise component coming from the jth receiver

ρ_j(t) is the thermal noise component coming from the jth receiver

If the coherent receiver has to work with the best performances, the LO power has to be much higher both of the incoming signal power and of the so-called “thermal” power. The thermal power is defined as the optical power that would cause a photocurrent with a power equal to the thermal noise power.

The second condition is generally easily realized, while the first could impose limitations on the amplification of the receiver preamplifier.

In these conditions, Equation 9.34 can be simplified as

(9.35)

Moreover, due to the fact that a complex electronic processing has to be performed after filtering, we will assume that all electronics functions will be performed via digital circuits. This requires a sampling of the baseband signal before filtering at a sufficiently high sampling rate. On the ground of the experience in electronic compensation, we will assume that five samples per pulse are sufficient to allow effective electronic processing and we will assume a sampling rate R_e ≈ 5R_S.

This means that in order to work at 100 Gbit/s with a 64-4QAM, the sampling rate has to be about 83 GHz, a very challenging speed, but probably within the reach of very high speed complementary metal–oxide–semiconductor (C-MOS) 22 nm application-specific integrated circuits (ASICS).

Increasing the number of levels, in order to transmit 100 Gbit/s using 512-4QAM with constant power (obtained by selecting 512 levels from the 600-4QAM reported in Table 9.4), the required sampling rate goes down to about 55 GHz, a speed of the order of that used in experimental systems with electronic nonlinear precompensation (experiments with sampling rates around 50 GHz are reported as an example in Chapter 6).

After baseband filtering, the four photocurrents are proportional to the quadratures of the field arriving at the receiver, but for the shot noise contribution.

A part the ASE noise, the quadratures of the received field, is related to the transmitted field quadratures by the fiber propagation characteristics causing chromatic dispersion, polarization rotation and PMD, and nonlinear distortions.

Let us assume in a first moment that fiber propagation is perfectly linear.

The role of the electronic postdetection processing is to compensate as much as possible linear propagation effects to recover the transmitted signal quadratures, to drive the frequency lock of the LO, and to estimate the transmitted symbol.

These functions can be recognized in the scheme of Figure 9.29b, where fiber-induced polarization rotations are compensated by multiplying the received quadratures by the inverse of the fiber polarization rotation matrix (the Jones matrix) in the Jones Matrix Inversion (JMI) block, and pulse distortion by residual chromatic dispersion and PMD are compensated via the Pulse Equalizer (PE).

The Jones matrix evolution at the first order (i.e., without the ω dependence that causes PMD) is tracked by the Jones Matrix Tracking (JMT) block, while the frequency lock loop driving signal is generated by the Frequency Control Circuit (FCC); finally the transmitted symbol is estimated by the Decision Circuit (DC).

The key elements of the electronic section of the receiver are the JMT and the FCC; electronic techniques to compensate linear pulse distortions are already reviewed in Chapter 6, and nothing more has to be added here for this particular case, while Decision Strategies in the general case of a multilevel signal will be discussed in Section 9.4.

9.3.3.2 M-DPSK Optimum Receiver

Although the quadrature receiver can work with M-DPSK signals once a suitable decision rule is implemented, it is not the optimum receiver, which is the receiver assuring the minimum error probability in the presence of optical noise.

Moreover, the quadrature receiver is very complicated in architecture and requires high speed electronics, with all the consequences in terms of costs and power consumption.

Thus, it is also practically very important to design a simpler receiver for M-DPSK signals.

The receiver scheme is shown in Figure 9.32 and is the analogous of the binary DPSK receiver shown in Chapter 6, but for the fact that to better use the received power a balanced configuration has been adopted.

The idea is to use the signal as a sort of LO by delaying it of a symbol interval and leveraging on the differential encoding at the transmitter.

If the transmitted signal has the form (9.16), at the receiver the signal writes as

(9.42)

where

k indicates the considered symbol interval

θ(k) depends on the symbol transmitted in the kth symbol interval

∈→ is the normalized field polarization vector

the ASE noise contribution is represented with η→ASE(t) (while it is to remember that η_T represents the function equal to one in the symbol interval and zero elsewhere)

The field on the upper branch of the receiver, after splitting, delay, and recombination, writes as

(9.43)

where α is the optical chain loss, and an analogous expression holds for the field in the lower branch and

(9.44)

Detecting the fields with a balanced receiver gives the following photocurrent

(9.45)

where n_s (t) is the shot noise term.

In equation (9.45) the symbol transmitted in the k-th symbol interval is represented by j, that determines the phase of the cosine function. The photocurrent is then sampled after low pass filtering and the samples are sent to the decision device.

The receiver we have described is much simpler than the whole quadrature receiver: no LO and no polarization tracking are needed due to the use of signal self-beating.

This is one of the reasons that qualify M-DPSK modulation as a good candidate for long haul 100 Gbit/s systems, where limiting the system cost is frequently much more important to push performances up to the limit.

9.3.3.3 M-PolSK Receivers

As in the case of M-DPSK, in the case of polarization modulation also, the quadrature receiver is not optimum and a direct detection receiver offering better performances can be designed [37,36].

The possibility to perform direct detection also contributes to simplify the receiver structure eliminating the need of the LO, but polarization tracking is needed since the transmitted information is coded in the polarization itself.

The idea on which optimum receivers, both coherent and direct detection, are designed is to detect the Stokes parameters to be able to operate processing and decision in the Stokes space.

The block scheme of the optimum M-PolSK receiver is presented in Figure 9.33, and it is composed of a front end producing four photocurrents that are proportional to the Stokes parameters and by an electronic processing section that is in charge to compensate polarization fluctuations, in the coherent case to drive the LO frequency locking and to perform symbol decision.

The block scheme of the coherent front end is reported in Figure 9.34; it is a combination of an heterodyne optical receiver and of square law type electronic demodulation that allows the Stokes parameters of the incoming field to be detected.

As far as Direct Detection is concerned, implementing an optical front end that is able to detect the Stokes parameters is equivalent to search the structure of a direct detection polarimeter. Several such polarimeters are used in practice, based on different principles, and the optical front end scheme of choice depends essentially on practical considerations.

An example of direct detection front end is reported in Figure 9.35: this structure is based on the analysis of the incoming field by means of quarter and half wave plates and polarization beam splitters. The input electrical field is divided by means of a polarization beam splitter into its linear polarization components and then these are combined by means of a couple of standard beam splitters of splitting ratio ε₁, a couple of standard beam splitters of splitting ratio ε₂, and a couple of polarization beam splitters as shown in the figure. If the optical path is accurately controlled, the two polarization components are combined in phase (technically means with the same phase) before detection on the second photodiode and in quadrature on the third one.

This structure is difficult to be realized via discrete components, due to the need of a very accurate control of the length of the light paths. On the other hand, it is quite suitable to be implemented by hybrid integration, where all the splitting structure shown in gray in the figure is integrated in a single silicon on silica chip. In this case, the phase control can be implemented by thermal phase controller and the structure results quite stable.

At the output of the interferometric polarimeter, the photocurrents results to be a linear combination of the Stokes parameters, thus inverting this relation, an estimate of the Stokes parameters is obtained.

Another interesting example of direct detection polarimeter is reported in [57] and it is based on the analysis of the incoming field by means of half and quarter wave plates. This structure does not require optical path length control and it is promising for discrete components implementation.

Finally, a polarimeter can be designed based on the fact that partially reflecting mirrors reflects the orthogonal linear polarization in different ways. Its schematic block diagram is shown in Figure 9.36.

Three photodetectors are coated with a reflection coating characterized by reflection parameters ψ_k, ρ_k, and κ_k (k = 1, 2, 3) while the fourth photodetector is antireflection coated so that all the optical power incident on it is detected. In particular, ρ_k indicates the reflectance of the surface for circular or unpolarized light, κ_k exp (i ψ_k) is the ratio between the complex reflection coefficient for linearly polarized light along parallel and perpendicular directions to the local incidence plane.

In all the cases we have considered up to now, a direct detection polarimeter produces four photocurrents that changes while the polarization of the received field changes. It is possible to demonstrate that in all the cases we have considered, there is a linear relation between the Stokes parameter and the currents at the output of the polarimeter that does not depend on the received polarization, thus it is possible to characterize the polarimeter with a matrix A so that, if the four currents at the polarimeter output are collected in a vector C, we can write [58,59]

(9.46)

Since the current at the output of the electrical demodulation in the coherent receiver shown in Figure 9.34 is also proportional to the Stokes parameters and the proportionality is a particular type of linear dependence, we will assume Equation 9.46 as valid for any type of optical front end in an M-PolSK receiver.

As far as the electrical processing section is concerned, it is in charge to perform both fiber polarization tracking and, in the coherent case, of generating the driving signal for the frequency lock of the LO.

Different algorithms have been proposed for these functions and we will not review them in detail.

Some algorithms use the same ideas we explained dealing with the quadrature receiver. In this case, the modulation is eliminated by using a decision driven algorithm that inverts the modulator operation starting from the estimate of the transmitted symbol [35].

Other algorithms use the idea that the constellation can be designed so that the average value of the Stokes vector over the entire constellation is not the null vector.

In this case, the modulation can be eliminated by simply averaging the modulated field over a time long with respect to the symbol time: the result is an average field whose polarization fluctuations are only due to the fiber propagation.

In this case, simply observing the variations of such an average vector, the fiber-induced polarization rotations can be compensated [35].

The last class of algorithms has the very important advantage not to require electronics speed comparable to the symbol rate, but it is not applicable to all the constellations, requiring an average Stokes vector different from zero.