Microwave photonics (MWP) is an interdisciplinary field that studies the interaction of microwave and optical wave [1,2] to realize generation [3–5], filtering and shaping [6–8], transportation [9,10], and measurement of broadband radio-frequency (RF) signals [11–15]. Recently, fueled by the rapid developments of photonic integration and advanced packaging technologies, more and more advanced MWP functionalities and systems are realized on photonic integrated chips. There are various material platforms available for integrated MWP (IMWP), such as silicon nitride (SiN) [16,17], indium phosphide (InP) [8,18], thin-film lithium niobate (TFLN) [19,20], and silicon-on-insulator (SOI) [21–24]. Among these platforms, the IMWP on SOI substrate is a competitive candidate thanks to its advantages in rich photonic device library, CMOS compatibility, high integration density, and potential of seamless integration with electronics [25]. Further, the imperfect opto-electronic properties of silicon can be resolved through hybrid or heterogeneous integration with diverse functional materials. As a result, many silicon-based IMWP systems have been successfully demonstrated with impressive performances, such as satellite communications [26], phased-array radars [27,28], and RF signal generating [29] and processing [30].

In an MWP system, the linearity is a crucial performance metric, which is usually quantified by the spurious-free dynamic range (SFDR) or the carrier-to-distortion ratio (CDR). It is known that an MWP system’s linearity is usually dominated by the electro-optical (EO) modulator [1]. Since the linear EO effect is lacking on silicon, most silicon modulators rely on the carrier-depletion effect to alter the refractive indices of silicon waveguides. Despite the merits in high operation speed and simple manufacturing process, the silicon carrier-depletion-based modulators are subjected to a complex modulation nonlinearity. Specifically, there are multiple sources responsible for the nonlinear distortion, i.e., the nonlinear modulation transfer function, the nonlinear capacitance of the reverse-biased PN junction, and the nonlinear free carrier dispersion (FCD) effect. These nonlinear mechanisms are coupled together, so it is challenging to linearize a silicon carrier-depletion-based modulator [31,32].

Our previous theoretical analysis in Ref. [33] shows that high-linearity silicon modulators can be realized by cascading dual Mach–Zehnder modulators (MZMs) either in series or parallel architectures. Under proper operation conditions, the individual nonlinear distortions of two sub-MZMs could cancel each other out to some extent. On the basis of the theoretical analysis, we demonstrate high-linearity dual-series and dual-parallel silicon MZMs in Refs. [34] and [35], respectively. For the dual-series MZM (DS-MZM), we tune the power-splitting ratio of the RF driving signal between the two sub-MZMs. The resultant SFDRs are $109.5,100.5{\mathrm{dB}·\mathrm{Hz}}^{2/3}$ at 1, 10 GHz [34]. Unlike the DS-MZM, the dual-parallel MZM (DP-MZM) configuration allows us to independently adjust the power-splitting ratios of RF and optical signals between the two sub-MZMs. Thanks to the added tunable variable, the SFDR is improved to $123,120{\mathrm{dB}·\mathrm{Hz}}^{6/7}$ at 1, 10 GHz [35]. In a more recent work [36], the nonlinearity of a dual-drive silicon single MZM is suppressed by independently tuning the reverse bias voltages and the RF driving signal amplitudes on two modulation arms. The corresponding SFDR is $123.4{\mathrm{dB}·\mathrm{Hz}}^{6/7}$ at 1 GHz.

The above research utilizes different architectures to substantially improve the linearities of silicon modulators. However, all these linearization schemes require one to tune the relative amplitudes of two RF driving signals precisely. Since an integrated broadband tunable RF power splitter is technically unfeasible at present, the previous works usually employ a 50/50 RF power splitter module and a bulky tunable RF attenuator module to adjust the amplitude ratio of two RF driving signals. The employment of tunable RF attenuator not only increases the cost and control complexity but also constrains the overall performances of the system in terms of the link gain and the RF bandwidth. Despite a lot of efforts, these techniques do not gain sufficient advantages for the deployment in practical systems. To address this issue, we propose an elegant scheme to linearize the silicon modulator in which the RF power tuning is replaced by the optical power tuning. Unlike the inconvenient RF power tuning subsystem [34–36], an integrated tunable optical power splitter is handy and possesses merits of unlimited RF bandwidth and high tuning resolution. The measured SFDR at 1 GHz is as high as $131{\mathrm{dB}·\mathrm{Hz}}^{6/7}$. To the best of our knowledge, this is the highest modulation linearity ever reached by integrated EO modulators. Further, the SFDRs at 10, 20, 30, and 40 GHz are 127, 118, 110, and $109{\mathrm{dB}·\mathrm{Hz}}^{6/7}$, respectively. This paper is organized as follows. First, the operation principle and the device design are discussed in Section 2. Subsequently, the performance of the proposed modulator is characterized systematically in Section 3. Finally, we compare our device with the recently reported high-linearity MZMs and then give an outlook about the additional linearization methods in Section 4.

A schematic diagram of the proposed silicon high-linearity MZM and its operation principle is shown in Fig. 1(a). It consists of two MZMs, which are cascaded in series through a $1\times 2$ optical switch (OS) based on the thermo-optic effect. The theoretical analysis in Ref. [33] manifests that the third-order nonlinearities of the two sub-MZMs can offset each other if their relative modulation depths satisfy a certain condition. In the previous experiment [34], the two sub-MZMs are driven by RF signals with different amplitudes and thus offer the desired modulation depths. However, it is inconvenient to precisely control the amplitudes of two RF signals in practical. To overcome this issue, we equally distribute the input RF signal between the two sub-MZMs, and then use the $1\times 2$ OS to manipulate the optical power imbalance between the two modulation arms of the second sub-MZM. In this manner, the ratio between the modulation depths of the two sub-MZMs can be adjusted freely without tuning the RF amplitudes.

The device is fabricated on a silicon-on-insulator (SOI) wafer with a 220 nm thick silicon layer and a 3 μm thick buried oxide (BOX) layer by Singapore AMF (Advance Micro Foundry). The two carrier-depletion-based sub-MZMs utilize 2 mm long single-drive series push–pull traveling-wave (TW) electrodes. A cross-section view of the sub-MZM is shown in Fig. 1(b), where the width and the slab height of the rib waveguide accommodating the PN junction are 500 and 90 nm, respectively. The two PN junctions embedded inside the modulation arms are connected serially in a back-to-back manner. This configuration reduces the capacitance loaded on the TW electrode by half and thus is beneficial to improve the modulation bandwidth [37]. The TW electrode is patterned on the top metallic interconnection layer, as shown in Fig. 1(b). The thickness of this layer is 2 μm. The signal line width of the TW electrode is 60 μm, while the gap between the signal and the ground lines is 40 μm. A 50 Ω on-chip TiN thin-film resistor is integrated at the end of the TW electrode to fulfill the impedance matching condition. The device contains three 200 μm long TiN-based heaters. Two of them are used to tune the bias points of the two sub-MZMs, while the third one is utilized to control the optical power splitting ratio of the $1\times 2$ OS. In order to avoid the RF crosstalk between the two TW electrodes, the pitch between the two sub-MZMs is set as 300 μm [38]. On the other hand, the intervals between thermal-optic phase shifters are $>800\mathrm{\mu m}$ to avoid the thermal crosstalk. This is helpful for stabilizing the operation state of the device. A microscope image of the silicon DS-MZM is illustrated in Fig. 1(c). Two fiber grating couplers (FCs) are used to interface the device with input and output single-mode fibers. The total fiber-to-fiber insertion loss of the device is 18 dB, of which $\sim 8\mathrm{dB}$ can be attributed to the two FCs and the rest 10 dB is caused by the DS-MZM.

By multiplying the modulation transfer function of the two sub-MZMs, we obtain the overall modulation transfer function of the DS-MZM, which can be written as $$\begin{gathered} I_{\text{out}}=\{\frac{{\left|E_{\text{in}}\right|}^2}{16}·\{\exp \left[-2\alpha (V_{b1}+\frac{v_{\mathrm{RF}}}{2\sqrt{2}})L\right]+\exp [-2\alpha \left(V_{b1}-\frac{v_{\mathrm{RF}}}{2\sqrt{2}}\right)L] \\ +2\exp \{-\left[\alpha \left(V_{b1}+\frac{v_{\mathrm{RF}}}{2\sqrt{2}}\right)+\alpha \left(V_{b1}-\frac{v_{\mathrm{RF}}}{2\sqrt{2}}\right)\right]L\}\cos (\mathrm{\Delta }{\theta }_1)\} \\ ·\{\left(1+\sin \phi \right)\exp [-2\alpha \left(V_{b2}+\frac{v_{\mathrm{RF}}}{2\sqrt{2}}\right)L] \\ +\left(1-\sin \phi \right)\exp [-2\alpha \left(V_{b2}-\frac{v_{\mathrm{RF}}}{2\sqrt{2}}\right)L] \\ +2\exp \{-\left[\alpha \left(V_{b2}+\frac{v_{\mathrm{RF}}}{2\sqrt{2}}\right)+\alpha \left(V_{b2}-\frac{v_{\mathrm{RF}}}{2\sqrt{2}}\right)\right]L\}\cos \left(\mathrm{\Delta }{\theta }_2\right)\cos \phi \}, \end{gathered}$$(1)where $E_{\mathrm{in}}$ represents the amplitude of input optical field, $\phi$ denotes the optical phase shift induced by the heater of the $1\times 2$ OS, and $\alpha (v)$ is the attenuation coefficient of the carrier-depletion-based modulation arm, which depends on the driving voltage $v$ falling on its PN junction. It can be translated into the waveguide propagation loss in the unit of dB/cm as $R_{\mathrm{loss}}={10·\log }_{10}[\exp (2\alpha ·1\mathrm{cm})]$. The DC reverse bias voltages on the two sub-MZMs’ PN junctions are represented by $V_{b1/2}$, while the optical path differences between two modulation arms of the two sub-MZMs are denoted as $\mathrm{\Delta }{\theta }_{1/2}$. For the single-drive series push–pull electrodes utilized here, $\mathrm{\Delta }{\theta }_{1/2}$ are calculated as $\mathrm{\Delta }{\theta }_{1/2}={\theta }_{\mathrm{bias}1/2}+2\pi L/\lambda [\mathrm{\Delta }{n}_{\mathrm{eff}}(V_{b1/2}+v_{\mathrm{RF}}/2^{1.5})-\mathrm{\Delta }{n}_{\mathrm{eff}}(V_{b1/2}-v_{\mathrm{RF}}/2^{1.5})]$. Here, ${\theta }_{\mathrm{bias}1/2}$ are the static bias phases of the two sub-MZMs produced by their heaters, while $\mathrm{\Delta }{n}_{\mathrm{eff}}(v)$ represents the effective index change of the carrier-depletion-based modulation arm. As aforementioned, the power of external RF modulating signal is divided equally among the two sub-MZMs. By combining the DC bias voltage and the AC modulating voltage, the driving voltages falling on the two modulation arms of the two sub-MZMs are $V_{b1/2}+v_{\mathrm{RF}}/2^{1.5}$ and $V_{b1/2}-v_{\mathrm{RF}}/2^{1.5}$. Here, $v_{\mathrm{RF}}$ denotes the external two-tone AC modulating signal, i.e., $v_{\mathrm{RF}}=v_0(\cos {\omega }_{1}t+\cos {\omega }_{2}t)$. In order to maximize their modulation efficiencies, both sub-MZMs operate at their quadrature points but with opposite polarities, i.e., ${\theta }_{\mathrm{bias}1}+2\pi L\mathrm{\Delta }{n}_{\mathrm{eff}}(V_{b1})/\lambda =\pi /2$ and ${\theta }_{\mathrm{bias}2}+2\pi L\mathrm{\Delta }{n}_{\mathrm{eff}}(V_{b2})/\lambda =-\pi /2$. If the propagation losses of the two phase-shifters are not taken into account, the modulation depth (MD) of the second sub-MZM under the bias condition described above is calculated as $$\begin{gathered} \mathrm{MD}=\frac{1+\sin \phi }{4}\{\begin{array}{c}\cos [\mathrm{\Delta }\theta \left(\frac{v_0}{2}\right)+{\theta }_{\mathrm{bias}}]\end{array} \\ \begin{array}{c}-\cos [\mathrm{\Delta }\theta (-\frac{v_0}{2})+{\theta }_{\mathrm{bias}}]\end{array}\}\mathrm{.} \end{gathered}$$(2)It is clear that different modulation depths can be realized easily by tuning the heating power of the OS.

Before calculating the modulation linearity of the DS-MZM with Eq. (1), we need to quantify the voltage dependences of the attenuation coefficient $\alpha (v)$ and the effective index change $\mathrm{\Delta }{n}_{\mathrm{eff}}(v)$. The DC modulation characteristics of the PN-junction embedded phase-shifter are obtained by measuring the transmission spectra of a reference asymmetrical MZM on the same chip. As shown in Fig. 2, discrete points present measured effective index changes $\mathrm{\Delta }{n}_{\mathrm{eff}}$ and propagation losses $R_{\mathrm{loss}}$ of the carrier-depletion-based phase shifter at different DC bias voltages. The nonlinear dependences of $\mathrm{\Delta }{n}_{\mathrm{eff}}$ and $R_{\mathrm{loss}}$ on the DC bias voltage $V_b$ can be fitted by fourth-order polynomials, i.e., $\mathrm{\Delta }{n}_{\mathrm{eff}}(V_b)=k_1·V_b+k_2·V_b^2+k_3·V_b^3+k_4·V_b^4$ and $R_{\mathrm{loss}}(V_b)=p_0+p_1·V_b+p_2·V_b^2+p_3·V_b^3+p_4·V_b^4$. Here, $V_b>0$ implies that the PN junction is reversely biased. Solid curves in Fig. 2 represent the data fitting results. The values of fitted polynomial coefficients are listed in the insets of Fig. 2.

Substituting the polynomials of $\mathrm{\Delta }{n}_{\mathrm{eff}}(v)$ and $\alpha (v)$ into Eq. (1) yields a closed-form expression of $I_{\mathrm{out}}$. According to the nonlinear modulation theory in Ref. [33], the amplitudes of the first harmonic (FH) and the third intermodulation distortion (IMD3) components in the modulated optical field are calculated as $C_{\mathrm{FH}}=I_{\mathrm{out}}^{(1)}(v_{\mathrm{RF}})·v_0$ and $C_{\mathrm{IMD}3}=I_{\mathrm{out}}^{(3)}(v_{\mathrm{RF}})·v_0^3/8+5I_{\mathrm{out}}^{(5)}(v_{\mathrm{RF}})·v_0^5/192$, respectively, under the small-signal approximation. Here, $I^{(1)}$, $I^{(3)}$, and $I^{(5)}$ denote the first, third, and fifth derivatives of $I_{\mathrm{out}}$ with respect to $v_{\mathrm{RF}}$. The modulation linearity is characterized by the CDR, which is defined as $\mathrm{CDR}={(C_{\mathrm{FH}}/C_{\mathrm{IMD}3})}^2$ [33].

The calculated contour maps of CDR as a function of the DC reverse bias voltages $V_{b1/2}$ on the two sub-MZMs are plotted in Fig. 3 for different power splitting ratios $\beta$ of the OS. In this calculation, the amplitude of the external two-tone AC modulating voltage is $v_0=0.1\mathrm{V}$ to meet the small-signal condition. Meanwhile, the two sub-MZMs are kept at the quadrature points of $\pm \pi /2$ by adjusting their heating powers. If $\beta$ equals 0.5, the optical powers inside the two arms of the second sub-MZM are balanced. Since the two sub-MZMs are biased at the opposite quadrature points, their modulation effects would completely counteract each other in this case. Therefore, we vary the value of $\beta$ from 0.6 to 0.9 in the calculations of Fig. 3. A proper power splitting ratio $\beta$ can be selected according to the following two criteria: (i) the maximal CDR value achieved; and (ii) the sensitivity of CDR to the two DC bias voltages $V_{b1/2}$. The latter is important in practical since it determines the stability and robustness of the linear modulation point. According to the two selection rules and the calculation results in Fig. 3(a), an optimal value range of $\beta$ is from 0.7 to 0.8.

To further evidence the linearity improvement, we theoretically calculate the optical transmissions of the DS-MZM and a reference single MZM as functions of the modulation voltage. Here, the modulation voltage denotes the voltage variation relative to the DC reverse bias voltage on the PN junction. The results are plotted in Fig. 4, where the modulation voltages have been normalized by the half-wave voltage $V_{\pi }$, which is measured to be 7 V. According to the CDR results in Fig. 3(c), the $V_{b1}$ and $V_{b2}$ are set to be 5.5 and 3.5 V in this calculation so that the DS-MZM operates at its optimal linear modulation point. The DC reverse bias voltage on the reference single MZM’s PN junction is 4 V. Apparently, the modulation transfer function of the DS-MZM shows a broadened linear modulation region as compared with the reference single MZM.

The small-frequency response of a stand-alone MZM, which is a copy of the sub-MZM shown in Fig. 1(c), is characterized with a light-wave component analyzer (Keysight N4373D). The measured EO S21 parameters at reverse-bias voltages of 0, 2, 4, and 6 V are displayed in Fig. 5(a). The corresponding 3 dB bandwidths are 36, 41, 47, and 51 GHz. Here, the capacitance ($C_{\mathrm{PN}}$) of the single PN junction is $\sim 4$ and 1.7 pF/cm at the reverse bias voltages of 0 and $-6\mathrm{V}$, while the series resistance ($R_{\mathrm{PN}}$) is $0.5\mathrm{\Omega }·\mathrm{cm}$. The resultant intrinsic bandwidths are 80 and 187 GHz, respectively. These values are in line with the results in Ref. [39]. However, our device is driven by the traveling wave electrode. According to the traveling wave modulation theory, the bandwidth of the traveling wave modulator is mainly determined by three factors, i.e., the RF loss, group velocity match between microwave and optical signals, and the impedance matching. A detailed description about the traveling wave electrode design of silicon MZM can be found in Ref. [39]. The measured switching characteristic of the $1\times 2$ OS is shown in Fig. 5(b). The corresponding thermal tuning efficiency is 30 mW per $\pi$ phase shift.

The two-tone test is implemented to characterize the SFDR of the device. The testing principle is illustrated in Fig. 1(a). An input two-tone ($f_1$ and $f_2$) RF signal with a spacing of 1 MHz is at first equally divided to two parts. The two RF signals are combined with two independent DC reverse bias voltages and then drive the two sub-MZMs. The modulation nonlinearity leads to unwanted IMD3 components ($2f_1-f_2$ and $2f_2-f_1$). They are close to the FH components ($f_1$ and $f_2$) and thus are difficult to filter out. The measurement setup is shown in Fig. 6(a). The output of an ultralow-noise laser source (ULN15PC) at 1550 nm is coupled into and out of the chip by two fiber grating couplers. The relative intensity noise of the laser is $-165\mathrm{dBc}/\mathrm{Hz}$. In order to avoid the optical nonlinear absorption inside the silicon waveguide, the laser power is set to 80 mW. The two-tone signal is generated by an RF source (Keysight E8267D), and then equally distributed between the two sub-MZMs by a 50/50 RF power splitter (Weinschel 1534 DC-40GHz). A two-channel voltage source provides DC reverse bias voltages for the PN junctions of the two sub-MZMs. The DC reverse bias voltages are combined with the AC modulating signals via two bias-tees (SHF BT 65A) and then are applied on the two sub-MZMs via a 67 GHz PGSGSGP probe. Another multichannel voltage source offers driving currents for the three heaters of the device. The output of the device is amplified by an erbium doped fiber amplifier (EDFA, Keopsys) to compensate the insertion loss. An optical bandpass filter (OBPF, WaveShaper 2000A) suppresses the amplified spontaneous emission (ASE) noise induced by the EDFA.

Two 1/99 optical power splitters and a dual-channel optical power meter are used to monitor the optical power levels at the points before and after the EDFA. The first monitoring point enables us to control the operation point of the DUT, while the second monitoring point is used to keep the output power of the EDFA at 10 dBm during the SFDR measurement. The optical-to-electrical conversion is performed by a 50 GHz photodiode with a responsivity of 0.75 A/W. A 90 GHz electrical spectrum analyzer (ESA, N9041B UXA) then acquires the powers of FH and IMD3 components in the RF output of the PD.

In almost all reports about the high-linearity modulators [34–37], the SFDRs are tested only at frequencies below 10 GHz. However, in ultrabroadband MWP systems, the frequencies of RF signals usually reach several tens of gigahertz [40]. Therefore, we extend the frequency range of SFDR measurement to 40 GHz. To the best of our knowledge, it is the first measurement of modulation linearity at such a high frequency. It is known that, besides the linearity of the modulator, the value of SFDR depends also on the noise floor (NF) of the full link. The NFs of our link at different frequencies are measured and shown in Figs. 6(b)–6(e). For frequencies below 20 GHz, the NF is $-163\mathrm{dBm}/\mathrm{Hz}$. It slightly rises to $-157\mathrm{dBm}/\mathrm{Hz}$ at 40 GHz.

In order to manifest the linearity improvement of the proposed DS-MZM, we at first characterize the linearity of the reference single MZM for performance comparison. Previous works have proven that the nonlinearity in the EO response of the carrier-depletion-based phase shifter can compensate partially with that of the sinusoidal modulation transfer function of the MZI by simply choosing a proper reverse bias. In this manner, the single MZM can be linearized to offer a high SFDR [34]. In the left column of Fig. 7, the best SFDRs of the reference single MZM at 1, 10, 20, 30, 40 GHz are measured to be $113,108,107,103,90{\mathrm{dB}·\mathrm{Hz}}^{2/3}$. The corresponding optimal reverse bias voltage is 3.7 V.

The SFDR of the DS-MZM is measured in the following steps. At first, we carefully tune the values of $V_{b1}$, $V_{b2}$, and $\beta$ to search the optimal linear modulation point with the highest CDR. In this process, the currents on the heaters of the two sub-MZMs are adjusted accordingly to maintain the two sub-MZMs at the quadrature transmission points of opposite polarities. After finding the linear modulation point, we sweep the output power of the RF source and then read out the RF powers of different frequency components in the ESA. Here, the maximum output power of our RF source is 25 dBm. The calibrated overall RF loss introduced by the $1\times 2$ RF power splitter, bias-tees, RF cables, RF adapters, and probes is $\sim 12\mathrm{dB}$. Therefore, the maximum RF power that can be delivered to the DUT is $\sim 13\mathrm{dBm}$ ($v_0=1.4\mathrm{V}$; $v_{\mathrm{pp}}=2.8\mathrm{V}$). Meanwhile, SFDR is usually required to be fitted in small signal region [1,6,18,19,41]. Based on the above arguments, the RF power of two-tone signal applied on the modulator varies from $-2$ to 13 dBm in our SFDR measurement. The measurement results at different frequencies are displayed in the right column of Fig. 7. Compared with the single MZM, the SFDRs of the DS-MZM are dramatically improved to 131, 127, 118, 110, and $109{\mathrm{dB}·\mathrm{Hz}}^{6/7}$ at 1, 10, 20, 30, and 40 GHz. The practical values of $V_{b1}$, $V_{b2}$, and $\beta$ at the optimal linear modulation point are 4.7 V, 2.3 V, and 0.7, respectively. At this optimal point, the optical insertion loss of the $1\times 2$ OS is $\sim 2\mathrm{dB}$ (including $\sim 1\mathrm{dB}$ inherent insertion loss and $\sim 1\mathrm{dB}$ bias point-induced loss), which then reduces the RF gain of the link by $\sim 4\mathrm{dB}$ according to the square-law detection. It is worthy to note that these values are basically consistent with the theoretical prediction in Fig. 3(c).

To investigate the impact of temperature fluctuation on the linearity, we place the DUT on a thermoelectric cooler (TEC) and then measure the SFDRs within a temperature range from 30°C to 60°C. The result is shown in Fig. 8(a), where the variation of SFDR (i.e., the blue curve) at 1 GHz frequency is less than $1{\mathrm{dB}·\mathrm{Hz}}^{6/7}$. However, the heating power of the $1\times 2$ thermal-optic OS is related to the ambient temperature. To maintain the optimal power splitting ratio of 0.7, the heating power varies from 20 to 50 mW.

In Fig. 8(b), we change the optical power incident on the PD by tuning the gain of EDFA and then measure the corresponding SFDRs at 1 GHz. During this measurement, the DS-MZM is kept at the optimal linear modulation point. The optical power fed into the PD is below 10 dBm to avoid the saturation of the output photocurrent. As we increase the optical power received by the PD, the powers of fundamental and IMD3 components are elevated accordingly. Their intercept point then shifts to a higher position [19]. Consequently, the SFDR value would increase as measured in Fig. 7(b), where a 1 dB increment in optical power improves the SFDR by 1.7 dB.

We investigate the dependence of the modulation linearity on the wavelength in Fig. 8(c). Since the two sub-MZMs utilize the symmetrical Mach–Zehnder interferometer (MZI) architecture, the relevant modulation characteristics including the linearity are insensitive to the wavelength in principle. However, the insertion losses of the FC and the $1\times 2\mathrm{MMI}$ are wavelength-dependent. If the wavelength deviates from the central wavelength of 1550 nm, the increase of optical loss would deteriorate the RF gain of the link. The SFDR value then drops, as shown in Fig. 8(c).

In conclusion, an ultrahigh linearity silicon modulator is achieved by serially cascading a modulation-depth-tunable MZM with a normal MZM. By co-optimizing the modulation depth and the reverse bias voltages on the PN junctions, our device reaches record-high linearities of 131, 127, 118, 110, and $109{\mathrm{dB}·\mathrm{Hz}}^{6/7}$ at 1, 10, 20, 30, and 40 GHz. In Table 1 our device is compared favorably with reported high-linearity EO modulators fabricated on mainstream photonic integration platforms. It is clear that our device achieves the highest linearity and the widest frequency range. In the fifth column of Table 1, we compare the strength variation ranges of the RF driving signals in these SFDR measurements. To enable a fair comparison, the RF signal strength is expressed in terms of $v_{\mathrm{pp}}/V_{\pi }$. Here, $v_{\mathrm{pp}}$ denotes the peak-to-peak value of the RF signal. We note that, in most researches, the SFDRs are tested in the small-signal region, i.e., $v_{\mathrm{pp}}/V_{\pi }<0.2$ as listed in Table 1. In contrast, the RF signal in our measurement has a much wider amplitude variation range that is beyond the small-signal region.Table 1.

Performance Comparison of Our Device with the Integrated High-Linearity Modulators Reported Recently

As summarized in Table 1, reported linearization techniques can be categorized into three types: (i) ring-assisted MZM to modify the modulation transfer curve [18,19,43]; (ii) single or combined MZMs driven by a pair of RF signals with finely tuned amplitudes [34–36,42]; and (iii) electrical predistortion [44]. As proof of principle, these techniques are utilized alone or jointly to realize plenty of high-linearity modulators. However, their applications in practical systems are restricted due to issues such as resonance-induced narrow optical bandwidth, difficulty in precise RF amplitude control, and special fabrication process. In contrast, our scheme possesses prominent advantages not only in ultrahigh linearity but also in control simplicity and robustness, wavelength independence, and CMOS compatibility.

We note that advanced fiber remoting links demand SFDR better than $120{\mathrm{dB}·\mathrm{Hz}}^{2/3}$ at frequencies above 30 GHz [45]. To achieve this goal, the DS-MZM can be further improved from the following two aspects. At first, the two fiber grating couplers can be replaced by low-loss edge couplers. In this way, the total fiber-to-fiber insertion loss can be reduced by at least 4 dB. As a result, gain and SFDR of the link can be substantially improved according to the result shown in Figs. 8(b) and 8(c). Second, the RF power splitter can be integrated monolithically with the silicon DS-MZM as in Ref. [38]. In additional to the benefit in higher integration level, this scheme helps to significantly reduce the RF transmission and coupling losses. The high-linearity DS-MZM in this work is fabricated on the SOI platform by using the FCD effect. It is known that the bandwidths of silicon modulators based on the FCD effect can hardly exceed 100 GHz. For MWP applications requiring RF frequencies up to a few hundred GHz, we suggest to use advanced material platforms, such as thin-film lithium niobate (TFLN) and EO polymer, on which integrated MZMs with ultrahigh bandwidths over 100 GHz have been successfully demonstrated [46–48]. The basic operation principle and the device structure design proposed in this work are applicable and then can be transferred readily to these material platforms with the linear EO effect. With all these measures, we believe that our device can fulfill the performance requirements of future ultrahigh dynamic range MWP systems.

Acknowledgment. The authors thank Dr. Bing Wei, Training Platform of Information and Microelectronic Engineering in the Polytechnic Institute of Zhejiang University.