1. Introduction

Silicon photonics has shown great potential for next generation high speed data transmission [1,2], due to its inherent compatibility with existing complementary metal-oxide-semiconductor (CMOS) technologies. Various passive and active devices have been demonstrated [3,4], such as couplers, splitters, switches, modulators, and photodetectors. And among these devices, silicon modulators constitute a key building block [5,6] in silicon photonics considering its broad applications in short-reach optical interconnects for data centers and supercomputers.

The electro-optic (EO) modulation in silicon devices is based on the plasma dispersion effect [7]. Mach-Zehnder modulator (MZM) structures are often preferred owing to their broad optical bandwidth, high thermal stability and robust fabrication tolerance. For high-speed modulation, travelling-wave electrodes (TWEs) are the proper candidate rather than the lumped electrodes, and TWE-based silicon modulators have been extensively investigated [8–14]. Many characteristics of the Mach-Zehnder modulators including bandwidth and modulation efficiency have reached excellent levels. Ideally, the travelling wave electrodes can be designed with common impedance (e.g. 50 Ohm) and with a matched termination. In reality, fabrication processes always introduce a multitude of structure and property variations that are difficult to control. Indeed, fabrication induced variation of device characteristics have raised much attention recently in silicon photonics. Several studies have paid attention to the effect of the fabrication variation on the performance of the devices [15–19]. Most of these studies are limited to passive devices, such as ring resonators and directional couplers. For active devices, while the variation of device performance parameters is easily measured, finding its source in the underlying structures is more difficult. Generally, active devices are much more complicated to analyze than passive devices in terms of fabrication variation. For an important active device—the Mach-Zehnder modulator, one frequently encountered issue is that the device performance is often substantially influenced by the impedance variation of its electrode and the accompanying driving signal reflection issue. As the impedances of the electrodes and the resistance of termination resistor (if integrated on chip) often vary and cause substantial signal reflection on traveling wave electrodes (with varying reflection magnitude), significant variation of optical modulation characteristics occurs. However, precisely what aspects of the modulation characteristics can be influenced by signal reflection on electrodes and how to quantify such influence remain an unexplored problem.

In this paper, we systematically analyze the correlations between the reflection of the driving signal and the output signal characteristics of silicon MZMs. The reflection on the electrodes is represented by the S₁₁ response. The output signal is characterized by a series of performance parameters, including the signal to noise ratio (SNR), extinction ratio (ER), root-mean-square (RMS) jitter, peak-to-peak (PP) jitter, and bit error rate (BER). The results indicate that there are strong correlations between the on-electrode reflection and the BER, SNR. As we shall see, other performance parameters show a wide range of dependence on the signal reflection on the electrodes. These diverse behaviors are analyzed and explained from the device physics perspective. Some relevant fabrication variation scenarios in the underlying structures are discussed, and potential approaches to mitigating the effects of such variations are suggested.

2. Device design and fabrication

The schematic view and microscopic image of the device are presented in Fig. 1. The silicon-on-insulator (SOI) wafer has a 3 µm thick buried oxide and a 220 nm thick top silicon. In addition, the ridge waveguides have a width of 500nm, a height of 220nm and a slab thickness of 90 nm. An asymmetric MZM is designed with a length difference of 20 µm between two arms, which results in a free spectral range (FSR) of about 30 nm. The length of the phase shifter is 2 mm. Two grating couplers are used to couple light in and out of the chip. Then two 1×2 multimode interference (MMI) structures are utilized for forming the Mach-Zehnder interferometer. A lateral PN junction is embedded as shown in the inset of Fig. 1(a), where the P and N doping regions are doped to around N_a= 4 × 10¹⁷ cm⁻³ and N_d= 2.5 × 10¹⁷ cm⁻³, respectively. The P++ and N++ doping regions are 1.25 µm away from the center of the rib waveguide for ohmic contact. Additionally, traveling wave electrodes with a pattern of GSGSG (G: ground, S: signal) have been designed with around 50 Ohm impedance, assisted by the co-planar waveguide design principle and finite element simulation [8,20]. The aluminum electrodes have a thickness of 0.53 µm, and widths of 28 µm and 160 µm for the S and G electrodes separated by a 5.5 µm gap. The matched resistances are integrated on chip to reduce the reflection of the radio frequency (RF) signal. The devices were fabricated in a CMOS foundry.

 

Fig. 1. (a) Schematic view of the MZM and the inset is the cross-section diagram of the PN junction. (b) Microscopic image of the MZM. (c) Typical spectrum of the MZM.

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3. Device characterization

The high-speed modulation performance of the MZM is measured with the setup shown in Fig. 2. A pseudorandom binary sequence (PRBS) signal with a pattern length of 2³¹−1 is generated by a bit error rate tester (BERT). The electrical signal is passed through a microwave power amplifier and a bias tee to obtain a 3.2 Vpp driving signal with 1.7 V reverse bias voltage (same bias as in subsequent S₁₁ measurement). The half-wave voltage V_π is found around 12 V through a separate testing device without the termination load (to avoid the integrated load interfering with V_π measurement). Then the signal is applied to the MZM via a 40 GHz GSG probe to characterize the performance of each arm of the MZM. A tunable laser is used as the input for the modulators with a proper wavelength, and the modulator works at the quadrature point. The output optical signal is amplified through an EDFA and the amplified spontaneous emission (ASE) noise is filtered by a tunable optical filter. Finally, the signal is fed into a commercial 50 GHz photodiode (PD) and captured by Agilent wide band digital oscilloscope. The power of the receiving signal is maintained around 9 dBm before being received by the PD. Figure 3 presents the results and the insets show 25 Gb/s eye diagrams of the typical devices. Additionally, other key performance parameters of these devices are measured by the oscilloscope, such as SNR and ER. As shown in Fig. 3, the modulation performance varies substantially. For example, for the device with the best BER, the SNR is 8.97 dB and the ER is 10.7 dB; whereas for the device with the worst BER, the SNR is only 6.44 dB and the ER is 4.6 dB, respectively. The mean value and standard deviation of the signal-to-noise ratio are 7.95 dB and 0.65 dB, respectively. The mean and standard deviation of the extinction ratio is 7.7 dB and 1.8 dB, respectively.

 

Fig. 2. Experimental setup of the measurement system used to characterize the MZM. (LD: laser diode; PC: polarization controller; DUT: device under test; EDFA: erbium-doped fiber amplifier; PD: photodetector; VNA: Vector Network Analyzer; BERT: bit error rate tester; AMP: amplifier; DCA: oscilloscope.). The lower-right inset schematically illustrates signal reflection on the electrode due to imperfect impedance matching at the termination.

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Fig. 3. Representative eye diagrams at 25Gb/s cases (from left to right: best, intermediate, worst). Unit for the horizontal axis: 8 ps/div; unit for the vertical axis. 28.8, 20, 15.7 mV/div (left, middle, right).

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The RF driving signal reflection on the electrodes of the MZMs is characterized by a Keysight vector network analyzer (VNA). Before measurement, the testing system is calibrated with an impedance standard substrate so that the responses of the cables and probe are de-embedded. The S₁₁ response of each GSG electrode of the MZMs is measured over various frequencies under the 1.7 V reverse bias voltages, and the representative results are shown in Fig. 4. Evidently, the S₁₁ response varies substantially from each other among the fabricated devices.

 

Fig. 4. Representative S₁₁ response. (Lowest: blue; intermediate: red; highest: yellow).

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4. Analysis

Joint analysis of S₁₁ and the performance parameter variation uncovers underlying relations between them. The S₁₁ response is a frequency-dependent quantity, which complicates the analysis. To simplify the initial analysis, S₁₁ is averaged over certain frequency ranges to obtain a quick view. Figure 5 shows that the relations between the BER, ER, RMS jitter and the mean S₁₁ response over three specific frequency ranges of 0∼19 GHz, 0∼25 GHz, 0∼40 GHz corresponding to 3 colors (note that the averaging of S₁₁ over frequency does not include a negligibly small frequency range (near DC) typically not accessible by a VNA, 0∼0.0003 GHz in this case). The data rate is fixed at 25Gb/s in all testing. The three frequency ranges represent the range containing most of signal power (roughly 0.75 times the data rate [21]), an intermediate range, and the full frequency range of our testing system. Data in Fig. 5 are measured from a number of devices (same design) on different dies fabricated in one batch. It can be seen that the trends of the three performance parameters with respect to the mean S₁₁ response are not the same, which indicates that the effects of the S₁₁ variation on the various modulation characteristics of the devices are diverse. There appears a clear trend between S₁₁ and BER. The mean BER (on log scale) is around 2 × 10⁻⁹. The relation between S₁₁ and ER in Fig. 5(b) shows somewhat larger random variation, but a rough trend is still discernible. As to the RMS jitter, it is almost hard to ascertain a clear trend with respect to S₁₁. In addition, when the S₁₁ response is averaged over different frequency ranges, the trend for the same parameter can vary to a certain degree. For example, the trends for the BER and ER at the frequency range of 0∼19 GHz are slightly clearer than the other two frequency ranges. Possibly, in higher frequency ranges, other types of noise smears the relation between S₁₁ and BER/ER. For the RMS jitter, the trends also differ somewhat in different frequency ranges, but it is difficult to compare. Note that while the BER gives a verdict of signal quality at the immediate output of the modulator, other parameters characterize various aspects of the signal quality that can be important for transmission over a distance under various transmission conditions.

 

Fig. 5. The relations between the (a) BER, (b) ER, (c) RMS jitter and the arithmetic mean of the S₁₁ response over three specific frequency ranges of 19 GHz, 25 GHz, 40 GHz.

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In order to systematically and quantitatively analyze the relation between the BER, SNR, ER, PP jitter, RMS jitter and the S₁₁ response, we introduce the correlation coefficients ρ(S₁₁, x) between S₁₁ and the device performance parameter x. To account for signal reflection contributions from various frequencies, the arithmetic mean of S₁₁ is obtained over different frequency ranges (all started from f≈0), then the correlation coefficients ρ(<S₁₁>, x) between the mean S₁₁ and various performance parameters are calculated as shown in Fig. 6. Here the correlation coefficient between two parameters is calculated via the standard form [22],

 

Fig. 6. The correlation coefficients between the BER, SNR, ER, PP jitter, RMS jitter and the arithmetic mean of the S₁₁ response over different frequency ranges.

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The behaviors of various parameters in Fig. 6 can roughly be classified into two categories, in which the magnitudes and trends of correlation coefficients are disparate. In the first category, the BER, SNR, and ER behave similarly with a high correlation coefficient (peaks >0.87). Their correlation coefficients (absolute value) all initially increase with the frequency span of averaging, reach a peak around 19 GHz, followed by a slight decrease with the frequency span and then a very slight increase. In the second category, both jitters (RMS and PP) behave similarly with a relatively low correlation coefficient (<0.8). As the frequency span increases, the correlation coefficients for the jitters first increase then decrease after a peak around 26 GHz (no increase at even larger frequency span). It is interesting to note that although Fig. 5(c) appears somewhat random, substantial correlation still exists between S₁₁ and the jitters. The difference of these two categories may be explained as follows. Obviously, the back-propagating signal on the electrodes due to reflection contributes to fluctuation of the overall driving signal on the electrodes, and hence fluctuation of the modulated optical signal level. (i.e. vertical signal level in the eye diagram in Fig. 3), it is straightforward to see that the SNR and ER are determined by the vertical signal levels, hence they show high correlation with S₁₁. The BER is also largely determined by the vertical signal level compared to a threshold, hence it also falls into the first category. The RMS jitter and PP jitter indicate the timing fluctuation of the signal, which characterizes the signal in the horizontal direction of the eye diagram. For an ideal abrupt step function-like transition, the jitter may not show any correlation with the vertical level fluctuation. In reality, the signal rises and falls in a gradual slope, along which the vertical signal noise translates into horizontal jitter. This may contribute to a concomitant correlation of the jitter with S₁₁. Note that the mean and standard deviation for the RMS jitter are 1.4 ps and 0.25ps, respectively, and for the PP jitter, 8.3 ps and 2.1 ps, respectively. Interestingly, for the ER, SNR and BER, the correlation peaks all appear around 19 GHz. Note that for 25Gb/s signal, most signal power is contained in the frequency range up to 19 GHz (roughly 0.75 of the data rate [21]). Note that the ER mainly reflects the ratio of the (mean) high level and the (mean) low level rather than the stability of them. For a modulator operating around the quadrature, if the output optical power has a linear dependence on the driving voltage, one can readily see that driving voltage fluctuation should have little influence on the mean level of “0” and “1” output states. However, in a silicon modulator with a p-n junction under reverse bias, the driving voltage causes the depletion region width to change nonlinearly ($W \propto \sqrt {{V_{bi}} - V} $ for a standard p-n junction, where V_bi is the built-in potential of the junction). This causes the total carrier numbers and the refractive index δn to vary nonlinearly with the driving voltage (note modulation output also nonlinearly depends on δn), hence on-electrode reflection-induced signal fluctuation will change the mean optical intensity of “0” and “1” states. This contributes to the correlation between S₁₁ and the ER. Some more complex fabrication variation scenarios may induce more complicated ER variation, and will be discussed in a later section.

As the frequency components of the driving signal are not uniform at all frequencies, it might be interesting to average S₁₁ by the weight of frequency components then re-examine the correlation. As shown in Fig. 7, the magnitudes of the correlation coefficients with the weighted mean of S₁₁ are comparable with the case of the arithmetic mean in Fig. 6. But the trends differ to some extent. When the average frequency range is 19 GHz, the absolute values of the correlation coefficients of the BER and SNR can reach 0.96 and 0.9 respectively. Unlike the case of the arithmetic mean, the correlation coefficients for SNR, ER, and BER in Fig. 7 tend to be stable when the average frequency ranges exceed ∼19 GHz, which is related to the low weight of frequency components at high frequencies. For the jitters, the correlation coefficients become stable after ∼23GHz. Note that if we average the arithmetic mean < S₁₁> over all devices, the mean is −16.3 dB with a standard deviation of 1.1 dB, respectively. If we do the same for the weighted mean, the mean and standard deviation are −14.8 dB and 0.76 dB, respectively. In subsequent analysis, we will use the weighted mean S₁₁ over 40GHz bandwidth for convenience (arithmetic mean yields similar results).

 

Fig. 7. The correlation coefficients between the BER, SNR, ER, PP jitter, RMS jitter and the weighted mean of the S₁₁ response over different frequency ranges.

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Note that the signal-to-noise ratio is related to the signal amplitude and noise (including reflection-related noise). In the measured devices, the amplitude of the modulated optical signal ΔP (difference of “1” and “0” states) also varies. To remove the influence of amplitude variation, one may use partial correlation coefficient p(S₁₁, x; ΔP) between S₁₁ and a modulated signal parameter x adjusted for ΔP (i.e. effectively keeping ΔP constant), where p is defined as [22]

All correlation coefficients ρ(a, b) in the formula of p(S₁₁, x; ΔP) can be readily calculated in a way similar to any correlation coefficients previously discussed. The partial correlation coefficient p(S₁₁, SNR; ΔP) = −0.85 remains high, but is a bit smaller than the corresponding ρ(S₁₁, SNR) (Table 1). Since the BER is often directly related to the SNR (see, for example, well-known formulas in Ref. [23]), the BER may also be affected by ΔP. The partial correlation coefficient p(S₁₁, BER; ΔP) remains very high as shown in Table 1. It is straightforward to see that ΔP can also affect the extinction ratio, thus the partial correlation coefficient p(S₁₁, ER; ΔP) is also computed and its absolute value is found smaller than |ρ(S₁₁, ER)| by a somewhat larger margin. Table 1 summarizes p(S₁₁, x; ΔP) and corresponding ρ(S₁₁, x) for the amplitude parameters x = SNR, ER and for BER. It would be interesting to explore the partial correlation between jitters and S₁₁ adjusted for ΔP. Note that different modulation amplitude ΔP may correspond to different transition slope due to the aforementioned nonlinearity of the modulator. As the slope differs, the mapping of the (vertical) signal noise to the horizontal jitter also differs, which contributes to the variation of jitter with varying ΔP. The partial correlation coefficients can help remove the influence of varying ΔP. We find p(S₁₁, jitter; ΔP) for the peak-to-peak jitter and rms jitter are p = 0.55 and 0.25, respectively, which are smaller than the corresponding ρ(S₁₁, jitter). The stronger correlation between the peak-to-peak jitter and S₁₁ (compared to that between rms jitter and S₁₁) appears to suggest that the on-electrode signal reflection primarily contributes to some of the largest jitters whereas other factors contribute to relatively small jitters (for the RMS jitter, the contributions of the largest jitters due to reflection are diluted due to averaging, leading to smaller correlation). In case there is an interest to fathom the low end of the correlation coefficient, we have also explored the correlation of the rise/fall time of the signal with the reflection. The correlation between the rise time and S₁₁ appears fairly low, ρ=0.096, but that between the fall time and S₁₁ is ρ=0.67. To remove the potential influence of varying ΔP, the partial correlation coefficients of the rise/fall time with S₁₁ adjusted for ΔP can be computed, which yields p = 0.16 and 0.35, respectively. Note that here the rise time (often going together with the fall time) is mainly intended to serve as a reference of a low-correlation case, and such low correlation phenomena in any research are usually difficult to interpret and sometimes neglected as the underlying processes are often rather weak or buried by other factors. In case there is an interest to understand the relatively weak correlation of the rise/fall time, here some possible explanation is offered. One readily sees from Fig. 2 inset that the reflected signal on the electrode may be superposed on the original driving signal, which can distort the transition edge nontrivially. Because the rise/fall time is measured at 20%-80% (or 10%-90%) points of the transition edge, such distortion can modify the rise/fall time even if the inherent full transition duration (0-100%) could be fixed. For example, the nonlinearity due to a standard p-n junction (discussed in the ER analysis) may induce a change of the depletion region width $\delta W \propto \delta V/\sqrt {{V_{bi}} - V} $, thus same amount of voltage fluctuation δV (due to reflection) can result in quite different δW for the “0” state and “1” state. As “0” and “1” states respond differently to δV, the rise (“0”→“1”) and fall (“1”→“0”) transitions may behave differently also, considering that the reflection of a transition edge (see Fig. 2 inset) comprises the residual signal from its past (especially the initial state of transition) rather than its final state. This may be a possible contributing factor for the difference between the (partial) correlation coefficient of the rise time and that of the fall time. The mean and standard deviation of the rise time is 20 ps and 2.0 ps, respectively, and for the fall time, 18 ps and 1.4 ps, respectively. Note that since the temporal parameters such as jitters and rise/fall time may be substantially affected by some other factors (e.g. carrier dynamics in the p-n diode [5,24,25]), the sheer values of (partial) correlation coefficients of the temporal parameters should be treated with care. The relative magnitudes of (partial) correlation coefficients for BER, SNR, ER, jitter, and fall/rise time are more relevant, as they reveal important information of the varying degrees of influence that the on-electrode reflection exerts on various characteristics of the modulation signal in this batch of devices. It is interesting to observe that the order of C_BER > C_SNR > C_ER > C_{pp jitter} > C_{fall time} > C_{rise time} (C = |ρ| or |p|) is the same for both the correlation coefficients and partial correlation coefficients in this batch of devices. Again, we note that while BER measures the overall signal quality at the modulator output, the other signal parameters studied here also describe important characteristics of a modulated signal, particularly when evaluating the qualification of this signal for long-distance transmissions or signal processing.

Table 1. Partial correlation coefficient p and correlation coefficient ρ

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5. Discussion

Note that the fabrication variation can cause two types of reflection on the electrode: (1) end reflection due to imperfect matching of the electrode impedance and the on-chip termination (the degree of imperfect matching varying device-to-device); (2) distributed reflection due to the inhomogeneity along the traveling wave electrode. It is well-known that the first type of reflection generally shows up as a sine-like spectral shape for S₁₁ (see, for example, Ref. [9]), which is also evident here in Fig. 4. The second type of reflection usually induces irregular spectral fluctuation/ripples, which appears small compared to the first type of reflection in the current devices as shown in Fig. 4. Under reasonable fabrication conditions, local inhomogeneity can be controlled reasonably well to reduce the second type of reflection to a low level. However, the first type of reflection is often harder to control without extensive process tuning and/or very high-end equipment. Note that one readily sees that the first type of reflection can induce signal level fluctuation (i.e. noise) and directly contributes to higher BER. For example, the signal level of a “1” bit may be shifted up or down (pseudo-randomly) by the end-reflected RF signal that carries the residual signal of prior bits (taking “0” and “1” values pseudo-randomly). This contributes to the (bit-to-bit) fluctuation of signal level of “1” bits. The “0” bits can be affected similarly. Of course, one readily sees that the second type of reflection can also affect BER, and the correlation analysis introduced in this work is also applicable to the case where such reflection is stronger. Note that the resistances of the termination resistors were estimated to vary around 54∼57 Ω based on DC measurement. The impedance of the TWE is designed by the standard method [10]. The characteristic impedance of the TWE itself is designed to be around 50Ω, whereas the loading of the diode and other parasitic elements reduces the total impedance by about 10 Ω (note that some significant source of fabrication variation tends to increase the total impedance, as discussed later). The designed total impedance generally varies little (within ∼2Ω) with frequency for frequency >1 GHz. Precise direct measurement of the frequency-dependent total impedance of each individual device is difficult with the presence of the on-chip termination.

Because the active devices in silicon photonics are fairly complicated, prior work has proposed a two-level analysis scheme for fabrication variation analysis, initially for optical switches [19]. One may divide a complicated device into several key structures, then study how the device performance is affected by such key structures (e.g. directional coupler in a switch, or the travelling wave electrode in a modulator here), which represents the device-level fabrication variation analysis. In some cases, it is also possible to further study how the performance of a key structure is affected by lower level elements (e.g. waveguides in directional couplers). Here we focused on the fabrication induced performance variation at the device level, where the traveling wave electrodes are the key structure in question. As the BER/SNR/ER show high correlation with S₁₁, this appears to capture the most crucial fabrication induced variation at the device level in this case. Extensive analysis indicates that the reflection on travelling wave electrodes may contribute to diverse correlation behavior with the variation of many performance parameters (such as SNR, ER, jitters). Even the fall time can show some correlation with reflection on electrodes. Potential contribution factors or mechanisms of these diverse correlation behaviors are explained from the device physics perspective.

In principle, one can build a theoretical model to further understand the influence of on-electrode reflection on the modulator performance. For example, in a simple device level model for Mach-Zehnder modulators [23], the output power may depend on the electro-optic phase shift $\Delta \phi $ as follows

In principle, it is possible to further explore fabrication-induced variation analysis at the structure level (i.e. within each electrode), finding how the variation of the on-electrode reflection is related to the variation of a large number of low-level structure parameters (including but not limited to width, thickness, and resistivity of electrodes and termination resistors, various parasitic parameters due to related structures such as the diode). In principle, the relation between S₁₁ and the low-level structure parameters can be obtained with the help of advanced device theory [8,10]. Because the large number of low-level parameters involved, a given variation of reflection could correspond to numerous possible variation scenarios in a huge parameter space spanned by these low-level parameters (while the termination resistor is relatively simple, high complexity can arise from the traveling wave electrode as we shall see). An exhaustive study is difficult. Here we illustrate a few possibly significant variation scenarios. Generally, large structure features are not easily susceptible to variation. For example, the metal linewidth variation (typically 0.1∼ 0.3 µm depending on the mean linewidth) is less likely to be an issue because the electrode size is fairly large. The estimated variation of the TWE impedance due to metal linewidth variation is less than 2%, which contributes very little to the variation of on-electrode reflection. In contrast, small features in the sub-micron optical waveguide are more susceptible to variation. For example, the edge location of the p and n doping regions may have a relatively large effect because the p-n junction depletion width W_pn is on the order of 0.1 µm. As the typical linewidth variation is also on the order of 0.1µm, the edges of p and n regions could shift by a relatively large amount. Particularly, the shrinkage of the p and/or n region width may possibly create a gap (i.e. an “intrinsic” region) between the p and n regions. This may substantially change the junction capacitance. In a TWE model [10], the total impedance of the transmission line is given by ${Z_0} = \sqrt {({R + i\omega L} )/({1/{Z_{diode}} + G + i\omega C} )} $, where R, L, G and C are the resistance, inductance, conductance and capacitance per unit length for the metal TWE and the Z_diode is the diode impedance per unit length. Hence the total impedance is influenced by the contribution from the diode such as its junction capacitance C_j [8,10]. For a p-i-n diode with an i-region width of W_i, the junction capacitance is modified to [26]

Also note that complex forms of the transfer function of the modulator may arise when fabrication variation is considered, which may contribute to large ER variation. For example, consider the overlapping p and n region case. If the p and n regions slightly overlap due to fabrication variation, it is possible that during the “1” to “0” transition, the depletion-region edge sweeps across the interface between the narrow overlapping region (lightly p-doped to N_a−N_d) to the fully p doped region (may be called “punch through”). Then the diode characteristics (including the junction width-voltage relation) changes suddenly on this interface, and the transfer function of the modulator may change more abrupt than usual when the voltage sweeps across this point. No matter in this case or in the gap case (i.e. a p-i-n diode), $\Delta \phi \; ({{V_j}} )$ or $\Delta {n_{eff}}({{V_j}} )$ becomes a more complex function (not in the simple square-root form) and the simple transfer function ∼$\cos \left( {g\sqrt {{V_{bi}} - {V_j}} - \Delta {\phi_0}} \right)$ no longer applies. Indeed, it would be better to separately consider the transfer functions for the gap case and the overlapping case, both will yield more complex (nonlinear) functions. The reflection-induced signal fluctuation on the electrode may thus also influence the output signal in a complex manner. The relatively large ER variation may also be partially attributed to these complex fabrication variation scenarios.

It would be of interest to reduce the modulation characteristics variation due to the on-electrode reflection variation. One can readily see that if better fabrication control (e.g. on the linewidths, thicknesses of various structures) can reduce the variation of structure parameters, the reflection variation will be less. Of course, this may require more costly fabrication equipment/process-tuning. Sometimes, intentional bias of the structure parameters from the ideal design might also be helpful (e.g. intentional increase of the termination resistor’s linewidth to reduce its mean resistance to the target value). Furthermore, one may consider the fabrication-tolerant device design. Here are a few suggestions/guidelines. (1) Creating a fabrication-tolerant p-i-n diode. As discussed above, a slight offset of the p or n region edges during fabrication may sometimes forms a p-i-n diode, inducing substantial variation of the total impedance of the transmission line. To overcome such variation, one may instead replace the p-n diode with a p-i-n diode with a relatively wide i-region. As shown in Fig. 8, as W_i increases, the variation of junction capacitance becomes smaller, hence the variation of the total impedance also decreases. If a fabrication control of δW_i/W_pn=0.5 can be achieved, then a p-i-n diode design satisfying W_i/W_pn >2.6 can ensure the capacitance variation <20% (total impedance variation <∼5%). If a wider i-region satisfying W_i/W_pn>5.3 can be accommodated in the design, we can even achieve capacitance variation <10% (total impedance variation <3%). Note that a sufficiently wide i-region also ensures that fabrication variation cannot cause the p and n regions to overlap. (2) Creating a fabrication-tolerant overlapping region for the p-n diode. Alternatively, one may keep the p-n diode and intentionally design the p and n regions to overlap by a substantial margin (e.g. 0.4 µm), so that the overlap region will not disappear due to fabrication variation and nor will it be depleted to “punch through” to the fully p-doped region. This will help avoid the complex variation effects arising from the depletion edge oscillating between the overlapping region and fully p-doped region. Of course, the incorporation of a p-i-n diode or a large overlapping region containing counter-doping may have other implications, and trade-off may be made according to specific needs. (3) Mitigating nonlinearity. The above analysis indicates that for some modulation parameters (e.g. ER), their variation is related to the reflection variation through nonlinearity in the modulation. Hence mitigating the nonlinearity may potentially help in this regard. Useful techniques for mitigating the nonlinearity can be found, for example, in Ref. [28]. Generally, in the transfer function $P(V )= {P_0} + {P_1}\textrm{cos}[{\Delta \phi (V )- \Delta {\phi_0}} ]$, both the cosine and $\Delta \phi (V )$ are nonlinear functions. It is possible to design the device so that the nonlinearity of the two functions roughly cancel each other. Note that $\Delta \phi (V )$ depends on the junction depletion region change in response to the applied voltage, which can be modified by the diode doping profile [26]. In principle, it is possible to control the doping profile (e.g. through multiple doping) so that $\Delta \phi (V )$ almost behaves like an arccos function (at least in a certain voltage range) so that the P(V) is almost a linear function. Of course, non-ideal effects and cost need to be considered in practice. Further detailed discussion is beyond the scope of this work.

 

Fig. 8. Junction capacitance variation for a p-i-n diode when the i-region width is subject to fabrication variation by an amount δW_i. C₀=C_pin(W_i) is the original p-i-n junction capacitance when δW_i=0.

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6. Conclusion

In this work, we have systematically studied the correlations between the reflection of the driving signal on travelling wave electrodes and the modulated signal characteristics of silicon Mach-Zehnder modulators. Driving signal reflection on electrodes is found to affect many performance parameters in diverse forms. The correlation coefficient is introduced to systematically and quantitatively evaluate the influence of reflection variation. By averaging S₁₁ over different frequency ranges, the BER, SNR, and ER show different frequency-range dependent correlation behavior with < S₁₁> from the jitters. Partial correlation coefficients can be further introduced to help remove the contribution of other factors. The relatively weak correlation of rise/fall time with S₁₁ is also included for reference. This work shows that correlation analysis may reveal rich information underlying the apparent complex performance variation. Some relevant fabrication variation scenarios in the underlying structures are discussed, and potential approaches to mitigating the effects of such variations are suggested. The approach and the findings herein may also help explore fabrication-induced variation in other active devices.