## Abstract

We characterize the impact of the modulator material on chirp, digital signal processing (DSP) algorithms and system-level performance in coherent digital optical links. We compare theoretically, in simulations and experimentally the lithium niobate (LiNbO_{3}), indium phosphide (InP) and silicon (Si) integrated platforms. Distortions to vector diagrams are traced back to modulation physics, and are interpreted as quadrature crosstalk. In a back-to-back BPSK setup with an RF drive signal amplitude of 1.5*V*_{π}, we measure chirp parameters α of ~0, 0.10 and 0.06 and error vector magnitude EVM_{RMS} of 5.3%, 9.4% and 10.6% with the LiNbO_{3}, InP and Si modulators respectively. Both α and EVM_{RMS} are found to scale with the RF signal amplitude. In simulations, using a polynomial fit over a sinusoidal fit when pre-compensating the Si modulator transfer function slightly improves EVM (−0.6%). We also show that Si-related distortions can impact the efficiency of symbol timing recovery. In conclusion, phase and attenuation distortions in InP and Si modulators deteriorate the overall performance in coherent links, and cannot be neglected for large RF signal amplitudes. These results will benefit the optical communications community.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Material platforms for integrated optical transceivers have their own inherent advantages and disadvantages. When choosing a platform for a specific application, obvious variables such as foundry/process access, wafer cost and product scalability are first factored in. Each integrated optical/opto-electronic device also varies in terms of yield, efficiency and other characteristics depending on the chosen platform. In deployed coherent systems, in-phase quadrature modulators (IQMs) typically have a nested structure containing two Mach-Zehnder modulators (MZMs) [1]. The three main material platforms currently available for large-scale production of such IQMs are Lithium Niobate (LiNbO_{3}), Indium Phosphide (InP) and Silicon (Si). Each offers distinct appealing features. From recent publications on commercialized/close to market IQMs for digital communications, LiNbO_{3} and InP modulators both allow for reasonable *V*_{π} (1.5-4 V) and insertion loss (3-5 dB), and high bandwidths (>35 GHz). Their main limitations are a relatively low number of dies per wafer (long LiNbO_{3} modulators, small InP wafers), and the need for thermo-electric cooling (TEC) for most InP devices. On the other hand, silicon modulators have higher *V*_{π} (5-6 V), higher insertion loss (6 + dB) and lower bandwidths (30-35 GHz). However, silicon photonics offers more compact devices in general, larger wafers and a mature fabrication process [1–3].

The linearity of the modulation mechanism, also called material linearity, is another characteristic varying significantly between platforms. It characterizes the slope of the index change ∆*n* versus the applied voltage *V*. InP and Si modulators exhibit a non-linear$\Delta n\text{\hspace{0.17em}}(V)$ relationship. In coherent systems, material non-linearity leads to chirp visualized as distorted transitions between symbols on the vector diagram - the IQ map showing the received samples before decision [4]. Since coherent links usually include dispersion compensation, one might assume that chirp resulting from material non-linearity is of little practical importance. But the more thorough analysis presented here suggests that IQ map distortions resulting from material non-linearity can adversely impact coherent systems regardless of dispersion. Specifically, when combined with the random sampling phase at the receiver, these distortions can deteriorate the performance of symbol timing recovery algorithms. Material non-linearity can thus ultimately increase the error vector magnitude (EVM) and deteriorate the bit error rate (BER). Although impairments due to the modulator material have been studied to some extent for specific devices [4–10], there is no study comparing, across the main commercial integrated platforms, the impact of these impairments on standardized coherent links (in terms of chirp and EVM) or on the performance of widespread coherent DSP algorithms. Moreover, there is little information available about their mitigation. This manuscript addresses these three topics.

In Section 2, we first review the theoretical basis of modulation in the LiNbO_{3}, InP and Si platforms, then compare the linearity of their respective MZM transfer function (TF) based on measurements and simulations. In Section 3, we simulate a complete coherent circuit in Lumerical Interconnect and show received vector diagrams versus the three aforementioned modulator materials. We compare and explain the distortions to inter-symbol transitions based on coherent modulation/demodulation principles, MZM TFs and material linearity. We then quantify the system-level penalty incurred from these distortions. We also experimentally study the system-level performance of LiNbO_{3}, InP and Si modulators for the back-to-back BPSK case, based on chirp parameter analysis and EVM measurements at low bitrate. In all comparisons, the RF signal amplitude relative to *V*_{π} is fixed. In Section 4, we implement and compare different non-linearity pre-compensation schemes in 16-QAM circuit simulations, and discuss their ability to compensate for material non-linearity. We also quantify the impact of the modulator material on the effectiveness of symbol timing recovery at the receiver end, using a well-known timing error detection algorithm.

## 2. Modulation linearity across platforms

#### 2.1 Review of modulation physics

We provide here for each material platform a condensed summary of the modulation physics on which are based most currently deployed commercial IQMs. Other structures and modulation approaches exist but are not explicitly described.

The LiNbO_{3} crystal is non-centrosymmetric, thus enabling the linear electro-optic effect, or Pockels effect [11]. The electro-optic coefficient${r}_{ij}$of LiNbO_{3} is a 3rd rank tensor which is usually shown as a 6x3 matrix. Its most important component,${r}_{33}$≈30.8 pm/V (wavelength-dependent), is used for modulator applications. Pockels effect can be pictured as a small change in the index ellipsoid surface, which is defined as [12]:

*x*,

*y*&

*z*directions. For an applied electric field

*E*

_{z}, the ellipsoid of LiNbO

_{3}is distorted as:

*V*) for LiNbO

_{3}. Absorption for this material is very small and constant in the middle of the C band.

InP modulators are III-V devices that rely primarily on the quantum-confined Stark effect (QCSE) in multi quantum well structures, consisting for example of alternating intrinsic layers of InP barriers and InGaAsP wells between *n* and *p*-doped contacts [13]. While both electro-absorption (E-A) and electro-refraction (E-R) effects are concurrent in the multi quantum well (MQW) structure and mathematically linked through the Kramers-Kronig relations, the latter effect is of primary importance in IQMs, since the phase of the light needs to be modulated, not just its intensity. The E-R operating wavelength is typically close to the band-edge resonance (< 150 nm) to leverage a larger ∆*n* amplitude. The relation$\Delta {n}_{InP}(E)$, where *E* is the applied field, is a complex non-linear function that can nevertheless be approximated as a quadratic relation due to the dominance of the quadratic QCSE [14]:

*k*is a constant. More closely matching data and accounting for other effects in MQW structures (like Pockels effect),$\Delta {n}_{InP}(E)$can also be expressed as a power law [15]:where a ≈10

^{−13}and b ≈1.6 (at 1550 nm) are empirically determined. Index modulation efficiency and residual absorption$\Delta {\alpha}_{InP}(E)$depend strongly on the relative proximity of the operating wavelength with the band-edge resonance. InP E-R modulators operated close to resonance feature much stronger index modulation than their LiNbO

_{3}counterpart, but also significant absorption (several dB/cm), which is again non-linear [13]. Hence, InP phase-shifters are typically much shorter than LiNbO

_{3}devices. For both these materials, an in-depth treatment of modulator physics is found in [16].

Silicon modulators rely on plasma dispersion effect, or carrier refraction [17,18], because the material’s centro-symmetry inhibits the linear Pockels effect. The modulator structure is then built around a pn junction driven in injection (positive bias) or depletion mode (negative bias). Index modulation is primarily function of the carrier concentration ∆*N* and ∆*P* (cm^{−3}) overlapping the optical mode, a relation summarized by Soref [19,20]:

*k*

_{1}≈-5.4x10

^{−22},

*k*

_{2}≈-1.53x10

^{−18},

*p*

_{1}≈1.011 and

*p*

_{2}≈0.838 at 1550 nm. An analogous expression exists for attenuation, the latter increasing with ∆

*N*and ∆

*P*(

*k*

_{1},

*k*

_{2}> 0). To find an expression for$\Delta {n}_{Si}(E)$and$\Delta {\alpha}_{Si}(E)$, we must express ∆

*N*and ∆

*P*as a function of the applied voltage

*V*(or analogously,

*E*x

*l*, where

*l*is the separation between the electrodes). We resort to the well-known approximations of a basic pn junction [17]:

*c*

_{1}and

*c*

_{2}are constants depending on the junction design (doping strength and geometry),

*d*is the distance from the depletion region edge and${N}_{D}$,${N}_{A}$are respectively the donor and acceptor dopant concentrations. Thus, carrier concentration varies linearly with

*d*but exponentially with

*V*. Furthermore, the depletion width, thus the modulation efficiency, also varies non-linearly with the bias point and

*V*. Hence, the global$\Delta {n}_{Si}(E)$relationship is, like for InP, non-linear and rather complicated. At best one can always use high-order polynomial fits, such as in [21,22]:where

*a*,

*b*and

*c*are again empirically determined constants.

In section 2.2, we link the modulation physics of the different material platforms to their respective DC phase shifter characteristics and MZM transfer function.

#### 2.2 DC characteristics

To enable direct comparison, but also for the later definition of simulated MZMs, C band optical phase$\phi \text{\hspace{0.17em}}(V)$and attenuation$\alpha \text{\hspace{0.17em}}(V)$curves are required for the non-linear materials.

For InP, they are extracted from measurements on a commercial 1.8 mm-long MZM, at 1.55 μm. The attenuation${\alpha}_{i}(V)$can be retrieved for both$i=\{1,2\}$arms separately by blocking one of them and measuring the power at the output. After$\alpha \text{\hspace{0.17em}}(V)$is de-embedded, the${\phi}_{i}(V)$curves can be obtained from the normalized MZM TF:

The silicon$\phi \text{\hspace{0.17em}}(V)$and$\alpha \text{\hspace{0.17em}}(V)$curves are computed starting with (6) and the analogous empirical attenuation expression. Measured pn junction dopant concentration data from IME Foundry is used. After solving for the optical mode distribution in the lateral pn junction waveguide and computing the effective index$\Delta {n}_{eff}(V)$using typical ridge dimensions (500 nm x 220 nm), we use an effective phase-shifter length${l}_{eff}$of 5 mm to get$\phi \text{\hspace{0.17em}}(V)$and$\alpha \text{\hspace{0.17em}}(V)$for a typical Si MZM arm. For,$\phi \text{\hspace{0.17em}}(V)$we use

The InP and Si phase shifters are compared to an ‘ideal’ material, defined with perfectly linear$\phi \text{\hspace{0.17em}}(V)$and no loss, i.e.$\alpha =0\in V$. Figure 1 shows the $\phi \text{\hspace{0.17em}}(V)$and$\alpha \text{\hspace{0.17em}}(V)$curves for the InP, Si and ideal phase shifters. For all material platforms,$\alpha \text{\hspace{0.17em}}(V)$excludes device insertion loss. The phase modulation efficiency of the ideal material is not relevant, since in the forthcoming simulations the drive voltage is normalized to *V*_{π}, and is set arbitrarily. For InP and Si, the observed phase non-linearity is of course a direct consequence of their respective non-linear $\Delta n\text{\hspace{0.17em}}(E)$behavior, defined in Section 2.1. For both phase shifters, the *V*_{π} and$\alpha \text{\hspace{0.17em}}(V)$characteristics are representative of published data [15,17,18,24]. We note that for InP, α increases for more negative values of *V*, whereas it decreases for Si. This opposite$\alpha \text{\hspace{0.17em}}(V)$pattern is explained by the different dominant attenuation mechanisms.

We perform a first assessment of the impact of the modulator material by simulating a simple dual-drive MZM circuit biased at null, shown in Fig. 2(a). For each platform, the two phase-shifters are defined by the [$\phi \text{\hspace{0.17em}}(V)$,$\alpha \text{\hspace{0.17em}}(V)$] curves of Fig. 1. The resulting DC TFs, i.e. output power versus static voltage in both MZM arms, are shown in Fig. 2(b) for low negative voltage. For a more explicit comparison across platforms, we plot for each one in Fig. 2(c) the normalized electric field amplitude *E* versus relative drive voltage, for differential drive around two arbitrary bias points. For both InP and Si modulators, deviation from an ‘ideal’ (pure sinusoid) transfer function is globally weak, and more important at shallow bias than at deeper bias. These TF distortions originate from the$\phi \text{\hspace{0.17em}}(V)$and$\alpha \text{\hspace{0.17em}}(V)$characteristics of the phase shifters. In particular, it can be seen in Fig. 1 that$\phi \text{\hspace{0.17em}}(V)$is more linear around −6 V than around −3 V, for both platforms.

We now inspect more carefully the −3 V bias plot in Fig. 2(c). At a drive voltage of -*V*_{π}/2, the InP curve is slightly below the Ideal curve, whereas the Si curve is clearly above it. The opposite is true at + *V*_{π}/2. Moreover, for silicon, within the ± *V*_{π} range distortions are maximal around ± *V*_{π}/2, and minimal at 0 V and ± *V*_{π} (the phase of maximum distortion for InP is harder to estimate).

We finally extrapolate these findings to the coherent link case: if using a 2*V*_{π} amplitude RF signal in a quadrature branch, and if analyzing the related symbol transitions on a vector diagram acquired with sufficient sampling rate:

- 1) At transit phases corresponding to voltages of +
*V*_{π}/2 and -*V*_{π}/2 (¼ and ¾ of the transit time assuming a linear$V(t)$driver), distortions should have opposite polarities for InP and Si. - 2) A Si modulator should undergo two successive distortion peaks of opposite polarity during a symbol transition, around +
*V*_{π}/2 and -*V*_{π}/2. Since the ‘ideal’ symbol transition is a straight line (for a linear phase material), these distortions should then be ‘s-shaped’.

These assumptions will be verified with simulations and measurements of complete coherent circuits, in Section 3.

## 3. Modulator material effects in coherent systems

#### 3.1 Simulations–Qualitative analysis

We simulate a complete coherent transceiver, comprising a dual nested MZM IQ modulator at the transmitter and a 90° hybrid followed by balanced photodiodes at the receiver. The circuit is shown in Fig. 3. The goal is a direct comparison of the effects of the modulator$\phi \text{\hspace{0.17em}}(V)$and$\alpha \text{\hspace{0.17em}}(V)$curves on the received vector diagrams. The analysis in the section is mostly qualitative; a quantitative analysis an experimental data are presented in Section 3.2 and Section 3.3, respectively.

In the circuit of Fig. 3, the IQM is in turn defined by each of the three [$\phi \text{\hspace{0.17em}}(V)$,$\alpha \text{\hspace{0.17em}}(V)$] curve sets of Fig. 1, representing the ideal, InP and Si platforms. All four phase shifters of the IQM are identical. All other circuit variables and impairments are fixed (unless otherwise stated) and aim at matching typical values, based on lab equipment specifications and measurements. The most important simulation parameters are listed in Table 1.

The transmission optical fiber and Erbium-doped fiber amplifier (EDFA) are disconnected to simulate back-to-back configuration. Simulations are run at 28 Gbaud BPSK/QPSK, and the sampling rate is set at 128 samples/symbol to capture enough detail in inter-symbol transitions. Specifically, the 10 ps rise and fall periods of the NRZ module each translate (at the chosen symbol rate) to 28% of the symbol duration, or to ~36 samples per symbol transition. BPSK is a logical modulation format for a first assessment of single-driver transitions, whereas QPSK format allows to visualize dual-driver transitions while preserving acceptable graphical clarity. Higher-order formats are harder to analyze graphically due the high number of overlapping transitions. Transmitter and receiver electrical bandwidths are set to 50 GHz to avoid having channel bandwidth limitations interfere with modulator linearity comparisons, or having to implement an equalizer. Raised-cosine pulse-shaping at the transmitter is disabled: the channel is memory-less and is not bandwidth-limited. Modulators are driven in push-pull mode, definedas${V}_{1}(t)+{V}_{2}(t)=\text{cte}$, where${V}_{1}(t)$ and${V}_{2}(t)$are the RF signals in both arms. In MZMs, push-pull operation theoretically prevents generation of chirp$\Delta v(t)$, defined as [25]:

The DC bias of both drivers is set to −3 V. To emphasize material non-linearity effects and compare them fairly across platforms, the RF swing is set to 2*V*_{π} for all of them. Voltage would of course be limited in practice: this more realistic case is studied experimentally in Section 3.3. Synchronization delay between the in-phase and quadrature components of the signal is retrieved from the timing of their respective electrical eyes after detection. Finally, the phase of the local oscillator is tuned manually by optimizing the rotation angle of the vector signal analyzer (VSA) vector diagram.

Received BPSK and QPSK vector diagrams are shown in Fig. 4(a) and Fig. 4(b), respectively. The ideal transmitter material shows almost perfectly linear transitions between symbols. The other two plots show distorted transitions, with the silicon plot showing obvious s-bends. The chirp peaks for single-driver (lateral or vertical) transitions occur approximately at the ¼ and ¾ of the inter-symbol transit time, as was predicted in Section 2.2. At these instants, the distortions have opposite polarities for InP and Si, which is also as expected. The diagonal QPSK transitions are noisier than lateral/vertical transitions presumably due to the simultaneous movement of the drivers in both quadratures.

In an alternative test (not shown) where attenuation is set to zero for Si, the constellation points move further apart due to the increased received power, but the distorted aspect of symbol transitions is unchanged. This means that the observed distortions are caused primarily by the non-linear$\phi \text{\hspace{0.17em}}(V),$since neither voltage-dependent optical attenuation, nor modulator imbalance, nor bandwidth limitations are present in this simulation.

IQ map distortions can also be linked to the modulator material impairments conceptually. If MZMs with linear$\phi \text{\hspace{0.17em}}(V)$materials are employed in the in-phase and quadrature branches of the IQM, and if voltage-dependent attenuation$\alpha \text{\hspace{0.17em}}(V)$is negligible, then the transfer function of the two MZMs is perfectly sinusoidal. In this case the total modulated optical signal ${S}_{ideal}$after the IQM can be written as:

*V*

_{π}can be described in terms of Fourier series expansions. Hence, for non-ideal modulator materials, a more appropriate formulation for (12) is:

*I, Q*) leads to quadrature crosstalk, seen as distorted IQ map transitions, or s-bends. However, due to the perturbative level of the deviations to perfectly sinusoidal transfer functions for InP and Si, the extent of this quadrature crosstalk (related to modulator material impairments) is limited. In other words, in (13), the relative weight of the${c}_{n}$and${p}_{n}$coefficients is much lesser than the weight of the${b}_{n}$and${q}_{n}$coefficients. Note that this Fourier series analogy stands for the practical case where the drive voltage is limited to ±

*V*

_{π}(the series is periodic outside this boundary, but not the actual TF due to attenuation).

Lastly, in separate QPSK simulation sets where transmitter and receiver bandwidths are equal to the symbol rate (28 GHz) and raised-cosine pulse-shaping is enabled, it is found that increasing the roll-off factor of the filter or lowering the RF drive amplitude allow for straighter transitions between symbols on the IQ map, for all platforms. Furthermore, lowering the effective bandwidth of the channel, i.e. decreasing modulator and photodetector bandwidths from 50 GHz, to 35 GHz, to 25 GHz while keeping the symbol rate at 28 GBaud, also improves the linearity of transitions. This is thought to originate from the reduced spectral content allowed through the receiver.

#### 3.2 Simulations – Quantitative analysis

We now investigate the performance penalty incurred from the modulator-related distortions described in the last section.

In a first QPSK simulation, we apply a constant white noise loading of 1E15 W/Hz with a Gaussian intensity distribution after the transmitter. This extra noise load is added because the back-to-back simulations reported in Section 3.1 are error-free. No booster EDFA or fiber is connected. For each modulator various performance metrics are retrieved from the simulation, in which again all other parameters are fixed. Results are shown in Table 2 for a 2*V*_{π} RF drive:

In Table 2, the stronger inherent voltage-dependent attenuation of silicon around −3 V bias (see Fig. 1) is largely responsible for the deterioration of the OSNR and the other performance metrics for this material. The OSNR is defined as the ratio of the signal power, integrated on the OSA over the signal bandwidth, over the noise power (over a 12.5 GHz bandwidth).

The above simulation set can be made more realistic, noting that in a real (long-haul) coherent system, a booster EDFA would increase the optical power out of the IQM to reach a standard launch power. Hence, in a second back-to-back QPSK simulation set, we enable this booster EDFA after the IQM (shown in Fig. 3). Since the range of optical powers out of the IQM are comprised within a 4 dBm range across modulators (−1.75 dBm to 2.25 dBm), we assume a fixed EDFA noise figure of 4 dB. The noise loading employed for the first simulation set is then coupled to the amplified signal. The EDFA gain is tuned to yield a final launch power systematically equal to 0 dBm across modulator platforms. Finally, the noise loading is varied to sweep the OSNR. The BER versus OSNR curves for each modulator and for various drive voltages are shown in Fig. 5.

When the RF drive is set to 2*V*_{π} for all modulators, the OSNR penalty at the HD-FEC threshold of 3.8 × 10-3 (versus the Ideal curve) is negligible for the InP modulator and ~0.1 dB for the Si modulator. This small performance difference between modulators is due to 1) the EDFA gain compensating the material-related voltage-dependent attenuation and 2) the noise loading largely masking the distortions due to material non-linearity.

However, if a practical upper limit of 2 V_{pp} is imposed on the RF drive voltage off the modulators, the OSNR penalty for silicon increases to ~0.8 dB (the full 2*V*_{π} swing for the Ideal and InP IQMs is <2 V_{pp} at −3 V bias). This larger penalty comes from the reduced extinction ratio incurred from the RF amplitude drop.

To summarize the OSNR and BER simulation results, it is clear that the impairments related to the modulator material (namely voltage-dependent attenuation and phase non-linearity) impact the performance of coherent systems. However, the extent of this impact depends largely on the relative drive voltage used at the transmitter and on the overall noise level in the system.

#### 3.3 Experiments

We now characterize experimentally the effect of the modulator material on vector diagrams and EVM in the C band, using a market optical modulation analyzer (OMA). A schematic ofthe circuit is shown in Fig. 6. We test commercial LiNbO_{3} and InP IQMs, and a research Si travelling-wave MZM (TW-MZM) designed in-house and fabricated at IME Singapore. The LiNbO_{3} IQM driver is limited to the linear regime of operation, and is used here as an experimental linearity benchmark. The three modulators have similar electro-optic bandwidths, in the 25-30 GHz range. The DC Vπ of the InP and Si devices are respectively 2.5 V and ~8V, and both devices are push-pull driven. All MZMs are internally biased at null, and the pn junction of the Si TW-MZM is reversed biased at −3 V DC. The 10 dBm local oscillator output of the OMA is the light source for the modulator to minimize phase noise. From specifications and measurements, the device insertion loss are typically 12 dB for the LiNbO_{3} IQM, <12 dB for the InP IQM and 14-15 dB for the Si MZM biased at −3 V. To standardize the received optical power at the OMA to 7 dBm (optimal operation point) across platforms, the EDFA pump power is varied between 58% and 78% of its maximum value. To generate the RF data, a 10 GHz clock signal is sent to a bit pattern generator (BPG). The BPG generates a PRBS^31-1 data stream through a tunable 450-900 mV output. The BPG jitter and rise time characteristics are similar to those reported in Table 1. The RF signal is sent to a 25 GHz, 25 dB amplifier and to a variable RF attenuator before reaching the modulator (except for the LiNbO_{3} IQM, which has its own amplifier). The OMA comprises a 63 GSa/s, 23 GHz bandwidth coherent receiver, and its demodulation capabilities include symbol timing recovery and adaptive channel equalization. For the three modulators independently, the equalizer is fixed manually when the EVM is minimized. The OMA Software interpolates between the measured samples (6.3 Sa/symbol at 10 Gbaud) to reach 10 points/symbol, and displays vector diagrams and constellations [27]. Finally, a digital communication analyzer (DCA) with electrical/optical inputs is used in parallel with the OMA to calibrate the RF signals and optimize bias.

Transmission in the experiment is back-to-back and the speed is 10 Gbit/s, again to minimize the impact of channel bandwidth limitations on linearity comparisons. At this speed, the RF *V*_{π} values are expected to be close to the DC *V*_{π} values. Hardware availability constraints unfortunately limited the modulation format to BPSK. Two sets of measurements are run with the LiNbO_{3}, InP and Si modulators, with RF signal amplitudes at the modulator of 1.5*V*_{π} (limited by the RF amplifier) and 0.75*V*_{π}, respectively. 10^{5} symbols are collected in each acquisition. Results are shown in Fig. 7.

The spurious-free dynamic range (SFDR) is a common modulator linearity metric in RF photonics/analog applications [21,24,28-31], sometimes also used in direct-detect digital communications [32]. Nevertheless, to quantify the impairments due to material non-linearity in the context of coherent digital links, we believe the chirp parameter α, defined as [33]:

where $\phi $ and*E*are the phase and amplitude of the optical signal, is a more explicit metric [4].The 𝛼 parameters are calculated directly from the vector diagrams of Fig. 7 using the most likely transition paths. For the InP and Si modulators, 𝛼 is a different function of the transition phase. For this reason, we report the maximum value$\left|{\text{\alpha}}_{max}\right|$reached over the transition, instead of reporting 𝛼 at a fixed phase or power (like in [4]). The absolute value is used to remove the ambiguity caused by ‘dual-path’ transitions, such as for the InP case in Fig. 7(a). At the system-level, EVM is chosen as the main figure or merit because it provides information about both amplitude and phase errors, and captures more detail about channel distortions than binary metrics like the bit error rate (BER) [34,35]. EVM captures the difference between actual and ideal symbol locations in the constellation, and is usually reported as a fraction of the square root of the ideal constellation average power. The EVM

_{RMS}(%) is defined as [36]:

*r*

^{th}symbol in a sequence,${s}_{ideal,r}$is the associated ideal constellation point, and

*N*is the total number of symbols in the sequence. In the experiment, EVM statistics are computed by the OMA Software after demodulation. The back-to-back configuration yielded error-free measurements (as computed directly by the OMA Software from EVM measurements) for all modulator platforms, hence we do not report these BER values here.

In Fig. 7, the transition patterns are similar to simulations of Fig. 4(a), but show with slightly less chirp due to the reduced RF swing (2*V*_{π} in simulations). Other small pattern discrepancies, notably for InP, are most likely due to design differences between the simulated and experimental devices as well as to non-idealities of the experimental setup itself, such as quadrature offset errors at the transmitter and/or receiver. In Fig. 7(a), at 1.5*V*_{π} RF signal amplitude, the measured chirp is negligible for the LiNbO_{3} IQM, whereas$\left|{\text{\alpha}}_{max}\right|$ = 0.06 for the Si modulator and$\left|{\text{\alpha}}_{max}\right|$ = 0.10 for the InP modulator. However, the measured Si EVM_{RMS} is worse than InP ( + 1.2%) and LiNbO_{3} ( + 5.3%). Thus, EVM is not a function of$\left|{\text{\alpha}}_{max}\right|$alone. The most likely reasons explaining the higher EVM_{RMS} for Si are its higher insertion loss (compensated with higher EDFA pump power), but more importantly - since the silicon traces are not significantly noisier than the traces of the other 2 modulators - its s-shaped chirp pattern, which potentially interferes with the timing recovery function of the OMA. Supporting this hypothesis is the widened aspect of symbol locations for the Si modulator. The relationship between modulator material non-idealities and timing recovery in coherent systems is further investigated in Section 4.

In Fig. 7(b), halving the drive voltage to 0.75*V*_{π} decreases chirp and EVM for all materials, and brings the EVM difference between Si and InP down to + 0.3% for Si. This is readily understood in regard of the Si MZM TF of Fig. 2(c), where the distortion peaks located at ± *V*_{π}/2 are only reached if a peak-to-peak swing >1*V*_{π} is used. The scenario depicted in Fig. 7(b), i.e. < 2${V}_{pp}$for InP and < 6${V}_{pp}$for Si, is more realistic for applications than the large RF drive amplitudes of Fig. 7(a), although for silicon further reduction of drive voltage is suitable. Overall, our measured α and EVM results are on the same range as other experimental back-to-back coherent results [4,37,38].

In other studies [5,18], it is argued that quasi-zero-chirp operation, or linear IQ map transitions, are possible for lateral pn junction Si MZMs driven in push-pull mode. In [18], it is explained that for this specific device, zero-chirp is enabled by the opposite behavior of$\phi \text{\hspace{0.17em}}(V)$and$\alpha \text{\hspace{0.17em}}(V).$ In other words, the distortion caused by free-carrier absorption (residual non-linear intensity modulation) would cancel the distortion caused by non-linear free-carrier refraction (phase modulation), since the former effect decreases with more negative voltage but the latter increases. Although this effect might be beneficial, our simulation and experimental results show that complete chirp cancellation does not occur with such a Si modulator. This cancelling effect being theoretically prohibited for InP MZMs, since both attenuation and phase increase with reverse bias, it might also contribute to the larger chirp measured with this material.

In summary, we have shown in Section 3 that non-linear$\phi \text{\hspace{0.17em}}(V)$modulators lead to distorted IQ map transitions, and that silicon modulators have slightly worse EVM figures compared to InP and LiNbO_{3} devices given the same relative RF signal amplitude. These effects are amplified with large RF signals. This can be of high practical importance if, for example, optimal signal-to-noise ratio (SNR) is dominant over maximum voltage when prioritizing the operational modulator criteria. Now, since most coherent links feature significant signal processing, Section 4 investigates more in depth the influence of the modulator material on key DSP algorithms, both on the transmitter and receiver sides.

## 4. Modulator material impairments and signal processing

#### 4.1 Non-linearity pre-compensation

For high-order formats such as 16-QAM and beyond, MZM non-linearity pre-compensation is usually deployed to spread constellation points equally. One can assume that the non-linearity due to the MZM structure itself is dominant, and therefore pre-distort the RF signal based on a reference sine function. While this is a logical approach for linear modulators like LiNbO_{3}, the procedure can somewhat be improved for non-ideal materials considering the MZM TFs of Fig. 2(c). In these cases, a high-order polynomial fit can be used instead of a sine function. With either fitting method, the methodology simply consists, for each non-compensated voltage point, in retrieving from the fit function a compensated voltage for which the output corresponds to the desired linearized output level at the original point. Alternatively, one can simply create a voltage look-up table graphically.

The sinusoidal and polynomial TF approximations are shown in Fig. 8(a) for Si. We compare the performance of the fits in 16-QAM simulations by inserting non-linearity pre-compensation blocks in the circuit of Fig. 3, following the driver blocks. Note that since the 90${}^{\circ}$ quadrature offset is perfect in simulation, there is no need (or potential gain) for joint pre-distortion, and electrical signals in the *I* and *Q* drivers can be treated separately. In each driver, two NRZ modules are summed to generate 4-level electrical signals. No additional pulse-shaping is performed at the transmitter, and no post-processing is applied at the receiver.

In a first set of simulations, noise, electrical bandwidth (at transmitter and receiver) and the other impairments listed in Table 1 are disabled to better visualize the pre-compensation effect. 28 Gbaud vector diagrams (2^{12} symbols) obtained with a 2*V*_{π} RF amplitude on the Si IQM are shown in Fig. 8(b). Whereas the four central constellation points are too close to the outer levels for the non-compensated case, the sine approximation slightly over-compensates the TF (the central points are now little too close to the center). This was expected from the inset of Fig. 8(a): the sine fit and the Linear target curves are on opposite sides of the actual Si MZM TF. Nevertheless, the sine approximation yields an acceptable constellation point spreading. When using a high-order polynomial fit instead, the inter-level spacing on the IQ map becomes perfectly uniform. Moreover, and interestingly, we see in Fig. 8(b) that equalizing the spreading of levels on the IQ map using non-linearity pre-compensation does *not* mitigate the s-shaped aspect of inter-symbol transitions, whatever the MZM TF fitting method is. In other words, a mere re-distribution of the applied voltage does not cancel the distortions to the MZM TF due to material non-linearity. Note that in Fig. 8(b) the higher sample density near the central constellation points is only due to the multiple transition paths crossing there.

In Fig. 8(c), we show EVM values retrieved from Interconnect for all modulator materials after the impairments listed in Table 1 are enabled in the previous 16-QAM simulation. Overall, the performance improvement due to a polynomial versus a sinusoidal fit is negligible for the ideal material and InP, and small yet noticeable for Si (−0.62% EVM). This can be explained by the relative deviation of the 3 material platforms from sinusoidal TFs in Fig. 2(c): the TF distortion is proportional to the potential improvement when fitting it with a polynomial. In general then, for non-linear materials having a MZM transfer function deviating substantially from a sinusoid, polynomial MZM TF fits should be employed. Also in Fig. 8(c), EVM is worst for Si versus the two other modulator platforms, independently of the compensation method. Since no post-processing is performed, this EVM penalty for silicon is very likely due to its$\phi \text{\hspace{0.17em}}(V)$and$\alpha \text{\hspace{0.17em}}(V)$characteristics only.

Due to the difficulty of pre-compensating for material non-linearity, the related IQ map distortions might impact the demodulation process after the coherent receiver, especially the symbol timing recovery stage. This is investigated next.

#### 4.2 Timing recovery and receiver-side equalization

Due to the random sampling phase at the receiver, acquisition of samples in the distortion peaks of symbol transitions for non-linear materials is possible, potentially having an impact on timing recovery.

The symbol timing phased-locked loop (PLL) in coherent demodulation may comprise the following building blocks: an interpolator, a timing error detector, a loop filter and a controller [39]. One of the well-known implementations of the timing error detector is known as the Gardner algorithm, which can be summarized for QPSK as [40]:

*r*

^{th}symbol and${y}_{I},{y}_{Q}$are the samples in each quadrature. The algorithm thus requires 2 samples per symbol. In theory, if there is no bit transition in the in- phase branch between the${(r-1)}^{th}$symbol and the${r}^{th}$symbol, then$[{y}_{I}(r)-{y}_{I}(r-1)]=0$and no error adjustment is coming from that branch at that time (same for${y}_{Q}$). However, as explained in Section 3.1, for modulators with non-linear$\phi \text{\hspace{0.17em}}(V)$ there is some level of crosstalk between the two quadratures. On the simulated time traces after detection of Fig. 9(a), this crosstalk is seen as glitches proportional to the non-linearity of the modulator. The glitches happen when the bit value on the branch shown does not change, but a bit transition is occurring simultaneously in the other branch. They are also visible on eye transitions, in Fig. 9(b). These glitches may parasitically contribute to$u\text{\hspace{0.17em}}(r).$In practice however, variations to$u\text{\hspace{0.17em}}(r)$are smoothed by the loop filter, and the controller does not necessarily update with every incoming error sample. Nevertheless, material non-linearity could still have a negative impact on the performance of the entire symbol timing PLL.

To verify this, we leverage the 28 Gbaud QPSK vector diagrams of Fig. 4(b), where 2*V*_{π} RF swings were used, circuit impairments were enabled and *N* = 2^{12} symbols were sampled at 128 samples/symbol for each modulator material. The methodology is as follows:

- 1) Import the Interconnect raw vector data (I, Q coordinates) to Matlab;
- 2) Down-sample data to two Sa/symbol at best phase, i.e. select one sample in the middle of each symbol + one halfway in-between. We assume no sampling frequency offset;
- 3) Run the PLL Matlab object on all
*N*down-sampled symbols. The PLL features all aforementioned building blocks, and the error detector is defined with (16). The PLL bandwidth normalized to the sampling rate is 0.01, and its damping factor is$\sqrt{2}/2;$ - 4) Plot the timing-corrected constellation and retrieve EVM
_{RMS}calculation. The first 10 symbols (loop transitory stabilization) are ignored; - 5) Repeat successively for the worst down-sampling phase, i.e. select samples ¼ of a symbol period away from those of step 3), and for each modulator material platform.

Results are shown in Fig. 10. The relative timing of the data samples entering the PLL impacts its performance, but it must be remembered that results show the extreme cases (in a real system, the receiver sampling phase is random). Moreover, we note for Si + 1.7% EVM_{RMS} versus the two other material platforms if the sampling phase is optimal, but this difference increases to + 2.6% at the worst theoretical sampling phase.

Thus, the symbol timing recovery algorithm tested here seems to be affected by the distortions due to the non-linear$\phi \text{\hspace{0.17em}}(V)$behavior of the Si IQM. To confirm this, we re-ran the QPSK simulations of Fig. 4(b) in Interconnect, but this time tuning the transmitter laser power to yield a constant launch power (after the IQM) for all modulator platforms, equal to that of the ideal modulator. After sending the Interconnect vector data through the timing recovery algorithm in Matlab, following again the above procedure, only small improvements versus results of Fig. 10(b) (<0.1% EVM_{RMS} for InP and <0.2% EVM_{RMS} for Si) were obtained, overall leaving again a significantly worst EVM performance for the Si modulator versus the other two. This is a confirmation that phase non-linearity-induced distortions in Si modulators, regardless of material-dependent attenuation, mitigate the performance of symbol timing recovery. However, as discussed in Section 3.2, at lower OSNR the noisy samples should blur this effect. Moreover, like in Section 3.3, we expect the EVM penalty for Si to be reduced proportionally to the RF drive voltage.

Finally, impairments due to non-linear$\phi \text{\hspace{0.17em}}(V)$can possibly be mitigated on the receiver side with equalization, provided that the sampling rate is sufficient. However, in Section 3.3, we found that optimizing the receiver equalizer in order to minimize EVM leaves well visible distortions on BPSK vector diagrams. Consequently, it might not be possible to minimize simultaneously EVM and material linearity distortions with that method. Furthermore, symbol timing recovery happens prior to equalization and will therefore still be affected by these distortions. Regarding physical means of compensating for silicon modulator non-linearities, it has been shown that specific choices of device length or bias point [24], [28] allow for some linearization. Other design-based approaches to compensate for MZM non-linearity in coherent links have also been reported [41].

## 5. Conclusion

We have reviewed the main modulation mechanisms pertaining to the LiNbO_{3}, InP and Si IQM platforms, highlighting their inherent differences in terms of optical phase linearity and attenuation versus voltage. We linked IQ map distortions in coherent links to MZM transfer function distortions originating from a non-ideal modulator material. We further interpreted these IQ map distortions as a limited form of crosstalk between the two quadratures of the IQ modulator. In back-to-back QPSK simulations where the RF drive was standardized to 2*V*_{π} across devices, we found that the OSNR penalty at the HD-FEC threshold versus the ideal-material modulator was negligible for InP and <0.1 dB for the Si modulator. This penalty increased to 0.8 dB for Si when its drive voltage was limited to 2 V_{pp}. We also compared LiNbO_{3}, InP and Si modulators driven with the same relative RF amplitudes in back-to-back BPSK measurements. At 1.5*V*_{π,} we measured chirp parameters of respectively ~0, 0.10 and 0.06 for the LiNbO_{3}, InP and Si modulators. The Si MZM yielded the worst EVM_{RMS}: + 1.2% versus InP and + 5.3% versus LiNbO_{3}. Halving the RF amplitude reduced α and EVM for each platform, and the gap between Si and the two other platforms.

We also investigated ways to compensate for the distortions due to the IQM material. When pre-distorting the RF signal based on sinusoidal and polynomial fits of the MZM transfer functions, it was found that using a polynomial fit is slightly beneficial to Si transmitters in terms of constellation point spreading and EVM (−0.6% in simulations versus a sine fit), but does not allow to correct the bends in symbol transitions due to a non-linear $\phi \text{\hspace{0.17em}}(V)$behavior.

This led us to investigate the impact of these distortions on symbol timing recovery at the receiver end. It was found that a bad receiver sampling phase amplifies the EVM penalty for Si versus the InP and LiNbO_{3} modulators (up to + 2.6% EVM_{RMS} for Si in the theoretical extreme case), due to sampling occurring in the highly distorted symbol transitions.

To conclude, it is clear from the analysis performed in this work that distortions related to the modulator material, i.e. phase non-linearity and attenuation, adversely impact digital coherent links, especially when large RF amplitudes (relative to 2*V*_{π}) are used. Therefore, these impairments should be considered when selecting an integrated platform for coherent transmitters targeting a specific application.

## Acknowledgments

We acknowledge the support of Lumentum LLC, Lumerical Solutions, Keysight Technologies, TeraXion, the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Québec – Nature et technologies (FRQNT).

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