For optical modulators used in microwave photonics (MWP) systems and telecommunications such as photonic microwave filters [1], photonic phased array antennae [2], analog-to-digital converters [3], [4], and advanced modulation formats [5], [6], one of the important metrics to measure their performances is the linearity, which is characterized by the spurious-free dynamic range (SFDR) [7] . Though Mach–Zehnder modulators (MZMs) based on Lithium Niobate (LiNbO₃) or III–V semiconductors have a higher SFDR (up to 121 dB·Hz^2/3 for the former [8] and 128 dB·Hz^2/3 for the latter [9]), they have a large footprint and are also difficult to integrate with electronic circuits. Optical modulators made on the silicon platform surpass them as they are compatible with the complementary metal oxide semiconductor (CMOS) techniques for low-cost fabrication and monolithic integration with microelectronics on one single chip. In fact, silicon modulators have been developing rapidly in recent years [10]. For example, carrier-depletion-based MZMs have been demonstrated with an operation speed up to 40 Gb/s [11]–​[18].
对于微波光子学 （MWP） 系统和电信中使用的光调制器，如光子微波滤波器 [1]、光子相控阵天线 [2]、模数转换器 [3]、 [4] 和高级调制格式 [5]、[6]，衡量其性能的重要指标之一是线性度，其特征是无杂散动态范围 （SFDR） [7].尽管基于铌酸锂 （LiNbO₃） 或 III-V 半导体的马赫-曾德尔调制器 （MZM） 具有更高的 SFDR（高达 121 dB·前者为^{Hz 2/3} [8] 和 128 dB·后者为^{Hz 2/3} [9]），它们具有较大的占用空间，也难以与电子电路集成。在硅平台上制造的光调制器超越了它们，因为它们与互补金属氧化物半导体 （CMOS） 技术兼容，可实现低成本制造，并在一个芯片上与微电子器件进行单片集成。事实上，硅调制器近年来发展迅速 [10]。例如，基于载流波耗尽的 MZM 已被证明具有高达 40 Gb/s 的运行速度 [11]–[18]。

In order to improve the linearity of silicon optical modulators, many methods have been proposed and demonstrated. Khilo et al. have demonstrated that, under the differential detection and push–pull drive scheme with a proper operation point for a certain phase shifter length, the SFDR for the second-order harmonic distortion (SFDR_SHD) and the SFDR for the third-order intermodulation distortion (SFDR_IMD) can be canceled and the linearity of silicon MZM can be theoretically larger than that of conventional LiNbO₃ MZM [19]. Recent research shows that by using differential drive, the linearity of silicon MZM can be improved to 82 dB·Hz^1/2 for SFDR_SHD and 97 dB·Hz^2/3 for SFDR_IMD [20]. Microring modulators have been demonstrated to exhibit a SFDR_IMD of 84 dB·Hz^2/3, but it is not suitable for wideband microwave systems due to the poor SFDR_SHD [21] , [22]. SFDR_IMD of 106 dB·Hz^2/3 has been achieved in ring-assisted MZMs [23], [24], because the phase response of the ring resonator cancels the nonlinearity in the MZI sinusoidal transfer function.
为了提高硅光调制器的线性度，已经提出并演示了许多方法。Khilo 等 人。已经证明，在差分检测和推挽驱动方案下，在一定移相器长度下具有适当的工作点，可以抵消二阶谐波失真 （SFDR_SHD） 和三阶交调失真 （SFDR_IMD） 的 SFDR，并且硅 MZM 的线性度理论上可以大于传统的 LiNbO₃ MZM [19].最近的研究表明，通过使用差分驱动，硅MZM的线性度可以提高到82 dB·SFDR_SHD 为^{Hz 1/2,97} dB·SFDR_IMD 为^{Hz 2/3}[20]。微环调制器已被证明表现出 84 dB·Hz^2/3，但由于 SFDR_SHD 较差，它不适用于宽带微波系统[21] ， [22]。SFDR_IMD 为 106 dB·在环形辅助 MZM [23]、 [24] 中已经实现了 Hz^2/3，因为环形谐振器的相位响应抵消了 MZI 正弦传递函数中的非线性。

In this paper, we present a silicon MZM with a single-drive push–pull traveling-wave electrode (TWE) configuration with improved linearity. The TWE is optimized to provide impedance match and flat electro-optic response. Compared with conventional differential drive, the single-drive scheme can more effectively reduce the second-order harmonic distortion due to the two auto-aligned push–pull signals from one RF feed. The SFDR _SHD is measured to be 85.9 dB·Hz^1/2 with 3.9 dB improvement over the previous best result [20] and the SFDR_IMD is measured to be 97.7 dB·Hz^2/3. Due to its high linearity, the modulator can generate 2, 3, 4, 5-level pulse amplitude modulation (PAM) signals at the symbol rate of 40 Gbaud/s and a 8-level PAM signal at the symbol rate of 25 Gbaud/s.
在本文中，我们提出了一种硅 MZM，该硅具有单驱动推挽行波电极 （TWE） 配置，线性度更高。TWE 经过优化，可提供阻抗匹配和平坦的电光响应。与传统的差动驱动相比，单驱动方案可以更有效地减少由于来自一个 RF 馈电的两个自动对准推挽信号而导致的二阶谐波失真。经测得的 SFDR _SHD 为 85.9 dB·Hz^1/2 比之前的最佳结果提高了 3.9 dB [20]，测得的 SFDR_IMD 为 97.7 dB·赫兹^2/3。由于其高线性度，调制器可以生成 2、3、4、5 级脉冲幅度调制 （PAM） 信号（符号率为 40 Gbaud/s）和 8 级脉冲幅度调制 （PAM） 信号（符号率为 25 Gbaud/s）。

Fig.1(a) shows the schematic structure of our silicon MZM. Compared to the differential drive configuration, the single-drive features low chirp, low capacitance (two junction capacitors connected in series), and simplified RF connection interface [25], [26]. The length difference of two arms in the asymmetric MZI is 90 μm. The 3.3-mm-long TWE uses a symmetric coplanar strip (CPS) structure in a ground-signal (GS) configuration with the two metal strips connected to the p⁺-doping regions outside the MZI arms where the RF signal is applied. A dc voltage (V_d) is applied to the middle n⁺ -doping region to set the two p-n junctions at the reverse-bias mode. The silicon waveguide is 500 nm wide and 220 nm high with an etched depth of 160 nm. The p-n junction is positioned in the middle of the silicon waveguide with doping concentrations of ∼4×1017cm−3 and ∼1×1018cm−3 for the p- and n-doping regions, respectively. Inverse tapers are used at the waveguide ends for optical input- and output-coupling.
图 1（a） 显示了我们的硅 MZM 的示意图结构。与 差分驱动配置，单驱动器具有低线性调频、低电容（两个结电容器 串联）和简化的射频连接接口 [25]、 [26]. 不对称马赫-曾德尔调制器中两条臂的长度差为 90 微米。3.3 mm 长的 TWE 在接地信号 （GS） 中使用对称共面带 （CPS） 结构 配置，其中两个金属条连接到马赫-曾德尔调制臂外的 p⁺ 掺杂区域，其中 施加 RF 信号。直流电压 （VD） 施加到中间的 n⁺ -掺杂区将两个 p-n 结设置为反向偏置模式。硅波导宽 500 nm， 高 220 nm，刻蚀深度为 160 nm。p-n 结位于硅的中间 波导的掺杂浓度 ∼4×1017cm−3 ∼1×1018cm−3 分别为 P 和 N 掺杂区域。反向锥度用于 用于光输入和输出耦合的波导端。

Fig. 1.  图 1.

(a) Schematic structure of the MZM. Inset shows the cross-section of the TWE and the circuit model. (b) Optical microscope image of the MZM.
（a） MZM 的示意图结构。插图显示了 TWE 的横截面和电路模型。（b） MZM 的光学显微镜图像。

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Fig. 1(b) shows the optical microscope image of the fabricated device. The p-n junction is segmented with a 1-μm-long striation un-doped in every 10 μm length to ensure that current flows only in the metal strips to reduce the RF loss. The TWE is made of aluminum metal strips with a width of 60 μm and a thickness of 1.5 μm. The gap separation between the signal and ground metal lines is 50 μm. The dc bias line is connected to the middle n⁺-doping region. In order to reduce the electromagnetic interference between the RF and dc signals, the dc line is designed as 10 μm wide and 2 mm long to act as an inductor to isolate them. As shown in Fig. 1 (a), the two PN junctions are connected in series with the dc voltage applied to the common cathode of the PN junctions to set the reverse bias at Vd. An RF drive signal VRF is applied onto one end of the TWE (the G and S metal lines). The other end is terminated with an external 50 Ω resistor. It should be noted that a common ground is used for both the RF signal and dc bias and hence the dc voltage drop is on both PN junctions. Assuming the PN junction capacitances are equal (Cdep1=Cdep2), then the RF voltage drop on each PN junction is VRF/2. Because the two PN junctions are connected back-to-back, the RF voltage drop has an opposite sign. Therefore, the entire voltage drop is −Vd+VRF/2 on the left junction and −Vd−VRF/2 on the right junction, resulting in push–pull modulation at the reverse bias voltage Vd.
 图 1（b） 显示了所制造设备的光学显微镜图像。 p-n 结每 10 μm 长度用一个 1 μm 长的未掺杂条纹分割，以确保 电流仅在金属条中流动，以减少射频损耗。TWE 由铝金属带制成，宽度为 60 μm 和 1.5 μm 的厚度。信号线和接地金属线之间的间隙间隔为 50 微米。直流偏置线连接到中间的 n⁺ 掺杂区域。为了减少 射频和直流信号之间的电磁干扰，直流线路设计为 10 μm 宽，并且 2 mm 长，用作电感器以隔离它们。如图 1 所示  （a） 将两个 PN 结串联，直流电压施加到 PN 结的公共阴极 将反向偏置设置为 Vd 。一个 RF 驱动器 信号 VRF 应用于 TWE（G 和 S 金属线）。另一端端接一个外部 50 Ω 电阻器。它应该是 注意到射频信号和直流偏置都使用公共接地，因此直流电压降位于两个 PN 上 结。假设 PN 结电容相等 (Cdep1=Cdep2) ，则每个 PN 结上的 RF 压降为 VRF/2 。因为两个 PN 结是相连的 背靠背，RF 电压降有一个相反的符号。因此，整个电压降位于 −Vd+VRF/2 左结点和 −Vd−VRF/2 右结点上，导致 反向偏置电压 Vd 下的推挽调制。

This section presents a mathematical analysis of the linearity of the MZM. We assume that the MZM is composed of two ideal multimode interference (MMI) 3-dB couplers. The output electric field can be expressed as
本节对 MZM 的线性度进行了数学分析。我们假设 MZM 由两个 理想的多模干扰 （MMI） 3 dB 耦合器。输出电场可以表示为

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查看源 \begin{equation} E_{\rm out} = \frac{{E_{\rm in} }}{2}\bigg[ {e^{ - a_A } e^{i\left({\phi _A + \Delta \phi } \right)} + e^{ - a_B } e^{i\phi _B } } \bigg] \end{equation}

Pout|QuadPout|Peak==Pin[14(e−2aA+e−2aB)−12e−(aA+aB)sin(ϕA−ϕB)]Pin[14(e−2aA+e−2aB)+12e−(aA+aB)cos(ϕA−ϕB)].(2)(3)

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查看源 \begin{eqnarray} {P_{\rm out} } |_{\rm Quad} &=& P_{\rm in} \left[ \frac{1}{4}({e^{ - 2a_A }+\, e^{ - 2a_B } }) \right. \nonumber\\ && \left.- \frac{1}{2}e^{ - ({a_A + a_B })} \sin ({\phi _A - \phi _B }) \right]\\ {P_{\rm out} } |_{\rm Peak} &=& P_{\rm in} \left[ \frac{1}{4}(e^{ - 2a_A } + e^{- 2a_B}) \right. \nonumber\\ &&\left. + \frac{1}{2}e^{ - ({a_A + a_B })} \cos ({\phi _A - \phi _B }) \right]. \end{eqnarray}

aA,B(V)ϕA,B(V)==a0+a1V+a2V2+a3V3φ1V+φ2V2+φ3V3.(4)(5)

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查看源 \begin{eqnarray} a_{A,B} (V) &=& a_0 + a_1 V + a_2 V^2 + a_3 V^3\\ \phi _{A,B} (V) &=& \varphi _1 V + \varphi _2 V^2 + \varphi _3 V^3 . \end{eqnarray}

We assume the top and bottom arms have identical responses to drive voltage in the ideal case. During the push–pull modulation, the top and bottom arms are subject to opposite drive voltages of V and −V, respectively, and thus we have:
我们假设在理想情况下，顶部和底部臂对驱动电压的响应相同。在 推挽调制，顶部和底部臂承受相反的驱动电压 V 和 −V，因此我们得到：

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查看源 \begin{eqnarray} a_A + a_B &=& 2a_0 + 2a_2 V^2\\ \phi _A - \phi _B &=& 2\varphi _1 V + 2\varphi _3 V^3 . \end{eqnarray}

Therefore, (2) and (3) can be rewritten as:
因此，（2） 和 （3） 可以改写为：

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查看源 \begin{eqnarray} P_{\rm out} |_{\rm Quad} &=& P_{\rm in} \left[ \frac{1}{4}\left(e^{ - 2({a_0 + a_1 V + a_2 V^2 + a_3 V^3 })}\right.\right. \nonumber\\ && \left.+ e^{ - 2({a_0 - a_1 V + a_2 V^2 - a_3 V^3 })} \right) \nonumber\\ && \left.- \frac{1}{2}e^{ - 2({a_0 + a_2 V^2 })} \sin ({2\varphi _1 V + 2\varphi _3 V^3 })\right] \\ P_{\rm out} |_{\rm Peak} &=& P_{\rm in} \left[ \frac{1}{4}\left (e^{ - 2({a_0 + a_1 V + a_2 V^2 + a_3 V^3 })}\right. \right.\nonumber\\ && \left.+ e^{ - 2({a_0 - a_1 V + a_2 V^2 - a_3 V^3 })} \right) \nonumber\\ &&+ \frac{1}{2}e^{ - 2({a_0 + a_2 V^2 })} \cos ({2\varphi _1 V + 2\varphi _3 V^3 }) \end{eqnarray}

From the above equations, we see that three factors contribute to the nonlinearity: (i) optical loss modulation in the active arms, (ii) cubic nonlinearity of the phase modulation, and (iii) MZI sinusoidal transfer function.
从上述方程中，我们可以看到三个因素导致了非线性：（i） 有源臂中的光损耗调制，（ii） 相位调制的三次非线性，以及 （iii） MZI 正弦传递函数。

The RF drive signal with two frequency tones can be written as V=V0cos(ω1t)+V0cos(ω2t). With Taylor expansion and Jacobi–Anger expansion, the second term in the bracket of Eq. (8) can be rewritten as
具有两个频率音的 RF 驱动信号可以写成 V=V0cos(ω1t)+V0cos(ω2t) 。随着 Taylor 的扩展和 Jacobi-Anger 展开式，方程 （8） 括号中的第二项可以改写为

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查看源 \begin{eqnarray} &&e^ - 2\left({a_0 + a_2 V^2 } \right) \sin \left({2\varphi _1 V + 2\varphi _3 V^3 } \right) \nonumber\\ &&\qquad= A\left[ {\cos ({2\omega _1 - \omega _2 })t + \cos ({2\omega _2 - \omega _1 })t} \right] \nonumber\\ &&\qquad\qquad+ B\left[ {\cos ({\omega _1 t}) + \cos ({\omega _2 t})} \right] \\ &&\qquad A = e^{ - 2a_0 } \left[ {({7a_2 V_0 ^2 - 2})J_1 (z)J_2 (z) - 3a_2 V_0 ^2 J_0 (z)J_1 (z)} \right. \nonumber\\ &&\qquad \qquad \left.+ 2a_2 V_0 ^2 J_0 (z)J_3 (z) - 3a_2 V_0 ^2 J_2 (z)J_3 (z) \right] \\ &&\qquad B = e^{ - 2a_0 } \left[ {({2 - 9a_2 V_0 ^2 })J_0 (z)J_1 (z) + 6a_2 V_0 ^2 J_1 (z)J_2 (z)} \right.\nonumber \\ &&\qquad\qquad \left. + a_2 V_0 ^2 J_0 (z)J_3 (z) \right] \end{eqnarray}

e−2(a0+a2V2)cos(2φ1V+2φ3V3)=C[cos(2ω1t)+cos(2ω2t)] C=e−2a0[4a2V20J21(z)−a2V20J20(z)−2a2V20J22(z)−2(1−2a2V20)J0(z)J2(z)](13)(14)

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查看源 \begin{eqnarray} & & {\rm e}^{ - 2\left({a_0 + a_2 V^2 } \right)} \cos \left({2\varphi _1 V + 2\varphi _3 V^3 } \right) \nonumber\\ & & \quad= C\left[ {\cos \left({2\omega _1 t} \right) + \cos \left({2\omega _2 t} \right)} \right]\\ &&\ C = e^{ - 2a_0 } \left[ {4a_2 V_0 ^2 J_1 ^2 \left(z \right) - a_2 V_0 ^2 J_0 ^2 \left(z \right) - 2a_2 V_0 ^2 J_2 ^2 \left(z \right)} \right. \nonumber\\ && \qquad \left. - 2\left({1 - 2a_2 V_0 ^2 } \right)J_0 \left(z \right)J_2 \left(z \right) \right] \end{eqnarray}

Fig. 2 shows the experimental setup to measure the SFDR and perform the PAM-N (N = 2, 3, 4, 5 and 8) modulation. Light from a tunable continuous wave (CW) laser first went through a polarization controller to set the transverse electric (TE) polarization and coupled to the MZM through an on-chip inverse taper. The modulated light was amplified by an erbium-doped fiber amplifier (EDFA) to compensate for MZM insertion loss and followed by a 3-nm bandwidth optical filter to suppress the amplified spontaneous emission (ASE) noise. The light was finally received by a 100 GHz bandwidth photodetector (u²t XPDV4120R). The responsivity of the photodetector is 0.5 A/W. For the SFDR measurement, the MZM was driven by a microwave signal consisting of two tones at frequencies 1005 and 1015 MHz generated by a microwave generator (R&S SMB100A). The microwave signal was applied to the TWE of the MZM via a 40 GHz bandwidth microwave GS probe. The other end of TWE was terminated with a 50 Ω resistor. The SFDR was obtained by measuring the fundamental, SHD and IMD components on an RF spectrum analyzer (R&S FSUP50). For the PAM modulation, a 20-GHz arbitrary waveform generator (AWG) from Keysight (M8195A) was used as the RF drive signal to modulate the MZM and the modulated signal was measured by a 33-GHz real-time digital signal analyzer (DSA) from Keysight (DSAX93204A).
 图 2 显示了测量 SFDR 和执行 PAM-N（N = 2、3、4、5 和 8）调制的实验设置。来自可调谐连续波 （CW） 激光器的光首先通过偏振控制器以设置横向电 （TE） 偏振，然后通过片上反向锥度耦合到 MZM。调制光由掺铒光纤放大器 （EDFA） 放大以补偿 MZM 插入损耗，然后由 3 nm 带宽的滤光片放大以抑制放大的自发发射 （ASE） 噪声。光最终被 100 GHz 带宽光电探测器 （u²t XPDV4120R） 接收。光电探测器的响应度为 0.5 A/W。对于 SFDR 测量，MZM 由微波发生器 （R&S SMB100A） 产生的频率为 1005 和 1015 MHz 的两个音调组成的微波信号驱动。微波信号通过 40 GHz 带宽微波 GS 探头施加到 MZM 的 TWE。TWE 的另一端端接一个 50 Ω 电阻器。SFDR 是通过在射频频谱分析仪 （R&S FSUP50） 上测量基波、SHD 和 IMD 分量获得的。对于 PAM 调制，使用 Keysight 的 20 GHz 任意波形发生器（AWG）（M8195A）作为射频驱动信号来调制 MZM，调制后的信号由 Keysight 的 33 GHz 实时数字信号分析仪（DSA）测量（DSAX93204A）。

Fig. 2.  图 2.

Experimental setup to perform the MZM nonlinearity measurement and PAM-N (N = 2, 3, 4, 5, and 8) modulation.
执行 MZM 非线性测量和 PAM-N （N = 2、3、4、5 和 8） 调制的实验装置。

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We first measured the MZM optical transmission spectra with the reverse bias applied either to the top or the bottom arms. Fig. 3(a) shows the typical spectra at 0 V and 6 V reverse biases. The spectra are all normalized to a passive straight waveguide. The on-chip insertion loss of the modulator is around 9 dB. The free spectral range (FSR) is 6.3 nm.
我们首先测量了 MZM 光传输光谱，将反向偏压施加到顶部或底部臂。 图 3（a） 显示了 0 V 和 6 V 反向偏置时的典型频谱。光谱都归一化为无源直线波导。调制器的片上插入损耗约为 9 dB。自由光谱范围 （FSR） 为 6.3 nm。

Fig. 3.  图 3.

(a) Optical transmission spectra at 0 V and 6 V reverse biases. (b) Phase shift of top and bottom arms versus reverse bias. (c) Relative loss coefficient change with reverse bias. The dots are measured data and lines are polynomial fitting curves. (d) Loss reduction of top and bottom arms versus reverse bias.
（a） 0 V 和 6 V 反向偏置下的光传输光谱。（b） 顶部和底部臂的相移与反向偏置的关系。（c） 反向偏置的相对损耗系数变化。点是测量数据，线条是多项式拟合曲线。（d） 顶部和底部臂的损耗减少与反向偏倚。

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From the spectral shift, we can get the relative phase shift (ϕ ) versus reverse bias as shown in Fig. 3(b). The small-signal π phase change voltage Vπ is obtained according to the following equation:
从频谱偏移中，我们可以得到相对相移 （ ϕ ） 与反向偏置的关系，如图 3（b） 所示。 小信号π相变电压 Vπ 根据以下公式获得：

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查看源 \begin{equation} V_\pi \left. {\frac{{d\phi }}{{dV}}} \right|_{V = V_d } = \pi . \end{equation}

Therefore, the modulation efficiencies of the top and bottom arms at Vd=0V are VπL=0.98V·cm and 1.23 V·cm, respectively.
因此，顶部和底部臂的调制效率为 Vd=0V 分别为 VπL=0.98V ·cm 和 1.23 V·cm。

One sees from Fig. 3(a) that the extinction ratio (ER) of the interference fringe increases when the bias is applied to the top arm while it decreases when the bias at the bottom arm. Thus, it is inferred that the top arm has a higher loss than the bottom one, i.e., aA>aB. The loss imbalance of the two arms is probably caused by the fabrication imperfection. The ER (in dB unit) of the interference fringe is expressed as
从图 3（a） 中可以看出，该 当偏置施加到顶部臂时，干涉条纹增加，而当底部偏置时，干涉条纹减少 手臂。因此，可以推断出顶部臂的损耗高于底部臂，即 aA>aB 。两臂的损失不平衡可能是 由制造缺陷引起。干涉条纹的 ER（以 dB 为单位）表示为

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查看源 \begin{equation} ER = 20\log _{10} \left({\frac{{{\mathop{\rm e}\nolimits} ^{ - a_B } + {\mathop{\rm e}\nolimits} ^{ - a_A } }}{{{\mathop{\rm e}\nolimits} ^{ - a_B } - {\mathop{\rm e}\nolimits} ^{ - a_A } }}} \right). \end{equation}

Therefore, we have  因此，我们有

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查看源 \begin{eqnarray} \frac{{{\mathop{\rm e}\nolimits} ^{ - a_B } + {\mathop{\rm e}\nolimits} ^{ - a_A } }}{{{\mathop{\rm e}\nolimits} ^{ - a_B } - {\mathop{\rm e}\nolimits} ^{ - a_A } }} &=& 10^{\frac{{\rm ER}}{{20}}} \equiv k\\ a_A - a_B &=& \ln \frac{{k + 1}}{{k - 1}}. \end{eqnarray}

From the measured ER change with bias on the top or bottom arm, we can get aA(V)−aB(0) or aA(0)−aB(V) as a function of reverse bias voltage V, respectively, as shown in Fig. 3(c).
从顶部或底部臂上偏置的测得的 ER 变化中，我们可以得到 aA(V)−aB(0) 或 aA(0)−aB(V) 作为反向偏置电压的函数 V，如图 3（c） 所示 。

The optical phase shift and loss versus voltage for each arm are fit with a third-order polynomial. The fitting parameters for each arm of the MZM are listed in Table I.
每个臂的光学相移和损耗与电压的关系与三阶多项式拟合。表 I 列出了 MZM 每个臂的拟合参数。

TABLE I Fitting Parameters for Each Arm of the MZM
表 I MZM 每个臂的拟合参数

We can convert the loss coefficient change into waveguide loss change by using the following formula:
我们可以使用以下公式将损耗系数变化转换为波导损耗变化：

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查看源 \begin{equation} \Delta Loss_{A,B} \left({dB} \right) = 20\left[ {a_{A,B} (V) - a_{B,A} (0)} \right]\log _{10} e. \end{equation}

Fig. 3(d) shows the loss reduction as a function of reverse bias for the top and bottom arms. The loss reduction rate is similar for the two arms.
 图 3（d） 显示了顶部和底部臂的反向偏置的损耗降低。两个组的损失减少率相似。

The TWE was optimized to achieve a large electro–electro (EE) bandwidth [26] . The measured S-parameters (S₂₁ and S₁₁) under various bias voltages are shown in Figs. 4(a) and 4(b). The microwave probe and coaxial cables were calibrated before measurement using the standard short-open-load-through (SOLT) method. The 6-dB EE bandwidth increases from 14.2 GHz at Vd=0V to 23 GHz at Vd=4V , as observed from the S₂₁ curves of the TWE. The microwave reflection at the entrance of the TWE due to impendence mismatch is represented by the EE S₁₁ curves. The measured S₁₁ is below −15 dB, suggesting very low reflection in the frequency range from 10 MHz to 26.5 GHz and hence good impedance match with 50 Ω. We also measured the small signal EO response of the modulator by using a lightwave component analyzer (LCA) which covers the frequency range from 100 MHz to 40 GHz. As shown in Fig. 4(c), the measured EO 3-dB bandwidth is 15 GHz at Vd=0V and increases to 32 GHz at Vd=6V. The EO response is relatively flat from 3 GHz and 13 GHz when Vd≥2V, which is useful for the wideband microwave photonics applications such as analog-to-digital converters.
TWE 经过优化以实现较大的电-电 （EE） 带宽 [26] 。在各种偏置电压下测得的 S 参数（S₂₁ 和 S₁₁）如 图 4（a） 和 4（b）。微波探头 同轴电缆在测量前使用标准短路开路负载直通 （SOLT） 方法进行校准。这 从 TWE 的 S₂₁ 曲线中观察到，6 dB EE 带宽从 14.2 GHz 增加到 Vd=0V 23 GHz Vd=4V 。TWE 入口处的微波反射是由于 系数失配由 EE S₁₁ 曲线表示。测得的 S₁₁ 如下 −15 dB，表明在 10 MHz 至 26.5 GHz 的频率范围内具有非常低的反射，因此 与 50 Ω 的良好阻抗匹配。我们还使用 光波元件分析仪 （LCA），覆盖 100 MHz 至 40 GHz 的频率范围。如  图 4（c），测得的 EO 3 dB 带宽在 15 GHz 时 Vd=0V 为 15 GHz，在 Vd=6V 。EO 的响应是 当 在 3 GHz 和 13 GHz 范围内相对平坦时 Vd≥2V ，这对于宽带微波光子学应用非常有用，例如 模数转换器。

Fig. 4.  图 4.

(a) EE transmission response S₂₁ of the TWE. (b) EE reflection response S₁₁ of the TWE. (c) EO transmission response S₂₁ of the MZM.
（a） 电子工程技术委员会的 EE 传输响应 S₂₁。（b） TWE 的 EE 反射响应 S₁₁。（c） MZM 的 EO 传输响应 S₂₁。

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In order to find the optimal operation point to achieve the highest SFDR, we measured the output power ratio between the fundamental tones and the distortions (F/SHD and F/IMD) versus wavelength as shown in Fig. 5(a). The power of the fundamental tone at each wavelength was set to –50 dBm. Both traces F/SHD and F/IMD have two peaks at the positive and negative slopes of the optical transmission spectrum close to the quadrature operation points. Because of the nonlinearity of free carrier absorption loss and the fact that loss and phase responses of the two arms are slightly different due to fabrication imperfections, the SHD cannot be fully cancelled when the modulator is biased at quadrature. The input laser wavelength was set at the quadrature point (1551.5 nm) for the following linearity measurement. The optical power received by the photodetector was 12 dBm and the photodetector current was 8.58 mA. The bias voltage was set to Vd=0V. We measured the output signals including fundamental, SHD, and IMD with 10 Hz resolution bandwidth and the noise floor observed from the RF spectrum analyzer was –146 dBm, mainly contributed by shot noise, thermal noise and source laser intensity noise. SFDR was obtained by measuring the fundamental, SHD, IMD tones when the power of two input tones varies, as shown in Fig. 5(b). Ten data points were used to fit the curves of the fundamental, SHD, and IMD. The fundamental tone has a linear response to the input power with a slope of 0.996. The SHD has a quadratic response to the input power with a slope of 1.997. The IMD power has a cubic response to the input power with a slope of 2.987. The noise floor is at –156 dBm with 1 Hz resolution bandwidth. The SFDR _SHD and SFDR_IMD for this single-drive push–pull MZM at 0 V bias are hence 85.9 ± 0.45 dB·Hz^1/2 and 97.7 ± 0.55 dB·Hz^2/3, respectively. Compared with the modulator with a differential drive configuration [20], a 3.9 dB improvement for SFDR_SHD has been achieved. The input intercept points (IIP2: 36.9 dBm, IIP3: 12.2 dBm) and the output intercept points (OIP2: 15.22 dBm, OIP3: –9.4 dBm) represent the input and output powers respectively when the fundamental tone curve intersects the SHD and IMD curves. We also measured the bias dependence of the SFDR. The inset of Fig. 5(b) shows that the SFDR_SHD is sensitive to the bias while the SFDR_IMD is almost independent of the bias [7].
为了找到实现最高 SFDR 的最佳工作点，我们测量了 基音和失真（F/SHD 和 F/IMD）与波长的关系，如下所示  图 5（a）.每个波长的基音功率设置为 –50 dBm。迹线 F/SHD 和 F/IMD 在光学器件的正斜率和负斜率处都有两个峰值 靠近正交操作点的传输光谱。由于自由载流子吸收的非线性 损耗以及由于制造原因，两个臂的损耗和相位响应略有不同的事实 缺陷时，当调制器在正交处偏置时，SHD 无法完全抵消。输入激光器 波长设置在正交点 （1551.5 nm） 处，用于以下线性度测量。光功率 光电探测器接收的电流为 12 dBm，光电探测器电流为 8.58 mA。偏置电压设置为 Vd=0V 。我们测量了产量 信号包括具有 10 Hz 分辨率带宽的基波、SHD 和 IMD，以及从 射频频谱分析仪为 –146 dBm，主要由散粒噪声、热噪声和源激光强度贡献 噪声。SFDR 是通过测量两个输入音的功率变化时的基频、SHD 和 IMD 音调获得的，因为 如图 5（b） 所示 。使用了 10 个数据点来拟合 fundamental、SHD 和 IMD。基音对输入功率具有线性响应，斜率为 0.996。这 SHD 对输入功率具有二次响应，斜率为 1.997。IMD 功率对输入具有三次响应 斜率为 2.987 的幂。本底噪声为 –156 dBm，分辨率带宽为 1 Hz。因此，在 0 V 偏置时，这种单驱动推挽式 MZM 的 SFDR _SHD 和 SFDR_IMD 为 85.9 ± 0.45 分贝·Hz^1/2 和 97.7 ± 0.55 dB·Hz^2/3 分别。比较 对于具有差分驱动配置的调制器 [20]，一个 SFDR_SHD 的改进达到了 3.9 dB。输入交调点 （IIP2： 36.9 dBm， IIP3： 12.2 dBm），输出交调点（OIP2：15.22 dBm，OIP3：–9.4 dBm）表示输入和输出 当 Fundamental Tone 曲线与 SHD 和 IMD 曲线相交时，分别幂。我们还测量了偏倚 SFDR 的依赖性。图 5（b） 的插图显示，SFDR_SHD 对偏置很敏感，而 SFDR_IMD 几乎独立于偏置 [7]。

Fig. 5.  图 5.

(a) Measured transmission spectra and the ratios of F/SHD and F/IMD as function of wavelength. (b) SFDR of the MZM at the quadrature operation point (1551.5 nm). The dots represent the measured data. The straight lines are the linear fitting lines. Inset shows the SFDR_IMD and SFDR_SHD measured at various bias voltages.
（a） 测得的透射光谱以及 F/SHD 和 F/IMD 的比率与波长的关系。（b） MZM 在正交工作点 （1551.5 nm） 的 SFDR。点代表测量数据。直线是线性拟合线。插图显示了在各种偏置电压下测得的 SFDR_IMD 和 SFDR_SHD。

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The high linearity property of the modulator can be exploited to generate PAM modulation signals. The input laser wavelength was set at the quadratic point to ensure the modulator was operated in the linear response regime. The RF drive signal from the AWG was amplified to a voltage swing of 7 Vpp. The bias voltage was set at 6 V. The bit error rate (BER) of the PAM signals as a function of optical signal-to-noise ratio (OSNR) was used to evaluate the modulation performance as shown in Fig. 6. Compared to a commercial LiNbO₃ modulator [27], this modulator shows about 3 dB OSNR penalty to achieve 1e-6 BER for 40 Gbaud/s PAM-2 modulation. Fig. 7(a) – 7(f) show the eye diagrams for PAM-N (N = 2, 3, 4, 5) modulations at symbol rates up to 40 Gbaud/s. The PAM-3 and PAM-4 modulations exhibit clear eye diagrams at 32 Gbaud/s yet slightly deteriorated at 40 Gbaud/s, partially limited by the low bandwidth of the DSA (33 GHz bandwidth). PAM-8 modulation was achieved at 25 Gbaud/s as shown in Fig. 6(f). A clearer eye-diagram could be obtained by using a higher bandwidth DSA assisted with post signal processing, such as pre-emphasis and equalization.
可以利用调制器的高线性度特性来产生 PAM 调制信号。输入激光器 波长设置为二次点，以确保调制器在线性响应模式下运行。俄罗斯联邦 来自 AWG 的驱动信号被放大到 7 的电压摆幅 Vpp 。偏置电压设置为 6 V。PAM 信号的误码率 （BER） 为 使用光信噪比 （OSNR） 的函数来评估调制性能，如下所示  图 6.与商用 LiNbO₃ 调制器相比 [27]，该调制器在实现 1e-6 BER 时表现出大约 3 dB 的 OSNR 损失 用于 40 Gbaud/s PAM-2 调制。 图 7（a） – 图7（f） 显示了 PAM-N （N = 2， 3， 4， 5） 调制的眼图 符号速率高达 40 Gbaud/s。PAM-3 和 PAM-4 调制在 32 Gbaud/s 时表现出清晰的眼图，但略有 在 40 Gbaud/s 时恶化，部分受到 DSA 的低带宽（33 GHz 带宽）的限制。PAM-8 系列 如图 6（f） 所示 ，在 25 Gbaud/s 时实现调制。一个 Clearer 可以通过使用更高带宽的 DSA 辅助后信号处理来获得眼图，例如 预加重和均衡。

Fig. 6.  图 6.

Measured BER as a function of OSNR. (a) The BER of PAM-3,4 at 32 Gbaud/s and the BER of PAM-8 at 25Gbaud/s. (b) The BER of PAM-3,4,5 at 40 Gbaud/s.
测量的 BER 作为 OSNR 的函数。（a） PAM-3,4 在 32 Gbaud/s 时的 BER 和 PAM-8 在 25Gbaud/s 时的 BER。（b） PAM-3,4,5 在 40 Gbaud/s 时的 BER。

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Fig. 7.  图 7.

Measured eye-diagrams of the PAM-N (N = 2,3,4,5, and 8) modulated optical signals.
测得的 PAM-N （N = 2、3、4、5 和 8） 调制光信号的眼图。

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We have presented experimental measurement on the linearity of a carrier-depletion-based silicon MZM with a single-drive push–pull drive configuration. The single-drive scheme can effectively reduce the second harmonic distortion due to the two strictly aligned push–pull signals from the one input RF feed. The MZM possesses a 3-dB EO bandwidth of 15 and 32 GHz at 0 and 6 V reverse biases, respectively. The SFDR_SHD and SFDR_IMD were measured to be 85.9 dB·Hz^1/2 and 97.7 dB·Hz^2/3 at the quadrature operation point, respectively. PAM modulation was realized using the high-linearity modulator. Eye-diagrams were measured for the PAM-2, 3, 4, and 5 signals at a symbol rate of 40 Gbaud/s and the PAM-8 signal at a symbol of 25 Gbaud/s.
我们提出了对具有单驱动推挽驱动配置的基于载流子耗尽的硅 MZM 的线性度的实验测量。单驱动方案可以有效减少由于来自一个输入 RF 馈电的两个严格对齐的推挽信号而导致的二次谐波失真。MZM 在 0 V 和 6 V 反向偏置时分别具有 15 GHz 和 32 GHz 的 3 dB EO 带宽。测得的 SFDR_SHD 和 SFDR_IMD 为 85.9 dB·Hz^1/2 和 97.7 dB·Hz^2/3 分别在正交操作点。PAM 调制是使用高线性度调制器实现的。以 40 Gbaud/s 的符号速率测量 PAM-2、3、4 和 5 信号，以 25 Gbaud/s 的符号测量 PAM-8 信号的眼图。