The bidirectional grating coupler, as the beam splitter/combiner of the hybrid electro-optic modulator, greatly affects the performance of the whole device [31–33]. The main parameters of the grating coupler include grating pitches, waveguide height, etching depth, duty cycles. When the single mode fiber (SMF) is perfectly aligned with the grating center, the light incident from the optical fiber is coupled and divided into two beams with the same amplitude and phase. Ideally, the BGC should feature low coupling loss, low parasitic reflections and large optical bandwidth. Here, we propose a grating design that fits the design requirements. Figure 1B shows a schematic diagram of the proposed silicon nitride BGC. The grating length is designed as 12 µm. To optimize the device performance, 2D and 3D Finite-Difference Time-Domain (FDTD) simulations were carried out [34]. A series of 2D simulations were carried out for rapid parameter optimization, the 3D simulations were carried out for the performance validation of the finalized design. By optimizing the grating structure parameters, the effective refractive index and coupling strength distribution in the grating plane can be optimized to find the best coupling performance.

For our proposed bidirectional grating coupler, which is both a perfect vertical grating coupler and a 3 dB beam splitter, in-plane optical coupling can be achieved through two access waveguides. Its design process is as follows. Firstly, the effective refractive index of the gratings is estimated by the following equation: N e f f g = F F ⋅ N e f f o + 1 − F F ⋅ N e f f e (1) where N _effo and N _effe are the effective indices of the original silicon nitride slab (400 nm-thick) and the etched areas (80 nm-thick), F F is the grating filling factor which is defined as the ratio between the length of the unetched section to the total period length of the grating. If FF of 0.5 is assumed, the effective refractive index of the gratings is calculated to be about 1.6. According to the Bragg condition, the grating period fit for peak wavelength of 1550 nm is about 968 nm. From the starting design of a uniform grating, we try to optimize the coupling performance by grating apodization. Thus, through a series of calculations using parameter scan and particle swarm algorithm, the optimized grating pitch length and the filling factor for different periods are fully explored. As shown in Figure 1B, only 6 grating periods (Λ _j , FF _j ) need to be designed due to the grating symmetry. Table 1 shows the optimized design parameters of the apodized grating.

Figures 2A, B shows the calculated optical field intensity distribution of in-plane and out-of-plane grating coupling respectively. As can be seen, this device can function well as a perfectly vertical grating coupler. Figure 2C shows the calculated performance of the designed silicon nitride bidirectional grating coupler, where the coupling efficiency (CE), the up-reflection towards fiber (UR) and the waveguide back-reflection (BR) spectra are all presented. The coupling efficiency reaches 70% at a wavelength of 1552 nm, implying that very close mode matching is achieved near the central coupling wavelength between the grating diffraction mode and the fiber mode. In addition, the waveguide back reflection is as low as -20.6 dB, and the minimum value of the upward-reflection to the fiber is 0.02%, corresponding to the return loss of -17 dB. The 1 dB bandwidth of the grating is 27 nm.

Vertical adiabatic mode conversion is an effective way for optical transmission between different waveguide layers in vertical direction [35]. For this device, optical adiabatic transmission is realized by the longitudinal evanescent wave coupling between inversed-taper silicon nitride waveguides and organic polymer waveguides, in which the alignment between the two waveguides is achieved by silicon dioxide cladding etching and organic polymer filling processes. The E-O material utilized in this device is 3:1 HD-BB-OH:YLD124, which has a refractive index of 1.77 near the wavelength of 1550 nm and optical absorption coefficient of 3 × 10⁻⁵ cm⁻¹. As depicted in Figure 1C, the structural parameters of the vertical mode converter (VMC) mainly include silicon dioxide spacer thickness H_CLAD and H_OX, silicon nitride waveguide height H_NW, silicon nitride etching depth H_NR, silicon nitride ridge waveguide width W_NR, polymer waveguide height H_PW, polymer waveguide ridge height H_PR, the polymer waveguide width W_PR.

The performance of the VMC is simulated using the EigenMode Expansion (EME) solver of Lumerical Mode solutions. To achieve high-efficiency coupling, three key device parameters should be carefully designed and optimized. The three parameters are the thickness of silicon dioxide spacer H_CLAD, the VMC length L and the tip width of the silicon nitride inversed taper respectively. After analyzing the influence of these parameters with a series of calculations, an optimized design with 99.2% transmission efficiency is obtained with a device length of 32 µm and tip width of 0.2 µm. Such a design is compact and easy to fabricate, and design values of other parameters are shown in Table 2. Figure 3B shows the calculated optical field distribution of the VMC and the optical mode profiles at different positions along the waveguide direction. Figures 4A, B shows the calculated normalized transmission of the vertical mode converter as a function of taper length and wavelength respectively. In the process, the silicon nitride waveguide and the organic polymer waveguide may cause additional errors due to the incomplete alignment.

We analyzed the transmission performance of the device when the center of the organic polymer waveguide deviates from the center of the silicon nitride waveguide, and obtained the results which is shown as Figure 4C. As the center of the organic polymer waveguide is shifted to both sides, the transmission performance will be reduced. However, in the range of about 250 nm for each deviation, the device can still maintain the transmission efficiency of more than 98%, and the insertion loss is less than 0.26 dB. Therefore, the device has a strong alignment tolerance.

Figure 5A is a cross section of the polymer optical waveguide, and Figure 5B is a top view of the modulator. The phase shifter is made up by E-O polymer waveguide and coplanar travelling-wave electrodes, where a single-drive push-pull operation scheme is utilized. The E-O material YLD124 combined with HD-BB-OH as the host polymer. The molecular structure of the material is shown in Figure 5B. According to Ref. [36], The electro-optic coefficient r₃₃ can reach the maximum value of 255 pm/V at the wavelength of 1550 nm. Here, the in-device r₃₃ of 150 pm/V is assumed and utilized in simulation. At a certain temperature, by applying a polarization voltage V_pol to the ground electrode, a polarization electric field can be formed in the ridge, and the chromophore dipoles in the electro-optic material are aligned along the polarization field, as indicated by the red arrows in Figures 5A, B. A driving microwave signal V_drive is applied on the signal electrode, and a strong electric field can be formed across the polymer waveguide. The modulation electric field of the left arm is antiparallel to the polarization direction of the chromophore, and the modulation electric field of the right arm is parallel to the polarization direction of the chromophore, as shown by the purple arrow in Figure 5A, thus realizing push-pull modulation features [37].

For this E-O polymer modulator, the modulation voltage can be expressed as the following equation [38], V π L = λ n 3 γ 33 Г d (2) where V _π is the modulation voltage applied to achieve a π phase shift, L is the phase shifter length, λ is the light wavelength in vacuum, d is the gap between the signal electrode and the ground electrode, and Г represents the E-O overlapping integral factor, which can be expressed as, Г = d V ⋅ ∬ E E x , y E o x , y 2 d A ∬ E o x , y 2 d A (3) where E _E (x,y) and E _O (x,y) represent the electric field profile induced by the applied voltage and the electric field distribution related to the optical modal profile of the polymer waveguide, A is the cross-sectional area of the polymer waveguide.

From the equation (2) and (3), it can be seen that the key to improve the modulation efficiency is to optimize the structure and relative location of the polymer waveguides and the coplanar travelling wave electrodes, when the E-O material is chosen. Normally, a better optical confinement of the polymer waveguide and a small electrode gap are beneficial to the E-O overlapping integral factor Г. However, decreasing the electrode gap will also lead to a larger overlapping between the electrode and the optical mode and thus a larger optical loss as a result. Therefore, the parameters of the phase shifter and the parameters of the electrodes can be optimized to achieve a compromise between the optical loss and the phase shift efficiency of the phase shifter. The optimization process is carried out using the Ansys Lumerical software package. Figure 6A shows the calculated modulation efficiency and optical loss of the phase shifter as a function of applied voltage and electrode spacing. As can be seen, the modulation efficiency and optical loss are all independent on the applied voltage, which is due to the linear modulation response of the Pockels effect. As a performance tradeoff, the electrode gap is designed as 4 μm, Г is 0.82, where a modulation efficiency of about 1.34 Vcm and optical loss of 9.48 dB/cm are obtained. The relative large optical loss is mainly due to the absorption of the OEO material. However, considering the polymer modulator is millimeter scale, the insertion loss is acceptable. The group refractive index of light wave is 1.81452 in optical simulation. In order to improve the bandwidth of the device, it is necessary to achieve rate matching and impedance matching. In addition, the microwave performance of the electrode is carefully analyzed and optimized for minimized microwave loss and impedance matching. After a series of calculations, all the design parameters are determined as shown in Table 3. Figure 6B shows the microwave loss and characteristic impedance of the travelling wave electrode as a function of microwave frequency. As can be seen, the microwave loss is lower than 14 dB/cm at the frequency of 100 GHz, and the characteristic impedance is around 50 Ω which shows a good impedance matching can be guaranteed. Figures 6C, D show the microwave refractive index and impedance at different signal electrode widths. With the decrease of the width of the signal electrode, the numeric matching between the group index of the microwave and the refractive index of the light wave is improved, which can achieve better rate matching. At the same time, the characteristic impedance is close to 50Ω.

The proposed silicon nitride/organic polymer hybrid modulator achieves a good balance between high modulation performance, low loss, and thermal stability. For ease of fiber packaging, silicon nitride bidirectional grating couplers are used as optical input/output interface and power splitter/combiner of the MZI modulator. Compared with silicon, silicon nitride exhibits the advantages of similar refractive index with E-O polymer [36], better thermal stability, large fabrication tolerance. To simulate the MZI modulator performance, all the optical components including the bidirectional grating couplers and the vertical mode converters are all modeled and simulated using FDTD or EME method, also S-parameter matrices are extracted for circuit-level simulation. The phase shifters are modeled and simulated with Lumerical Mode solutions and Charge, and then the optimized results and parameters are exported for circuit-level simulation.

Figure 7 shows the simulation link diagram of the modulator. The optical network analyzer (ONA) is adopted to obtain the static transmission spectra of the optical modulator, while a PRBS generator, a symbol mapper and a digital communication analyzer (DCA) are utilized for feeding PAM4 driving signals and eye diagram observation respectively. The length of modulator is chosen to be 3 mm. After setting the static bias conditions of the modulator and the wavelength of the laser, the static spectrum of the hybrid electro-optic modulator is obtained, as shown in Figure 8. Both balanced-arm MZI and unbalanced-arm MZI modulators are designed and simulated respectively, in a push-pull driving scheme. In Figure 8, it is shown that their half-wave voltages are both 2.29 V. Therefore, the modulation efficiency of the push-pull modulator is 0.67 V cm. Figure 9 shows the bandwidth of the modulator at different lengths. It can be seen the device length have a large impact on the bandwidth. In the simulation, the effect of the microwave loss, velocity matching and impedance matching of the device are all considered. Different microwave refractive index and impedance results have been obtained in Figures 6C, D, and the corresponding bandwidth is shown in Figure 9B. With a device length of 3 mm and signal electrode width of 28 μm, a bandwidth of 174 GHz can be obtained.

Linearity is an important metric when simulating PAM4 eye diagrams. When the linearity is poor, the three eyes of the PAM4 output eye diagram have different heights, and the bit error rate of the eye diagram determines by the smallest eye. R_LM is defined in IEEE 802.3bs to evaluate the linearity of the PAM4 eye diagram. V_A, V_B, V_C and V_D respectively represent the four level values from top to bottom in PAM4 eye diagram. The calculation formula is as follows: S min = 1 2 min V D − V C , V C − V B , V B − V A (4) R L M = 6 ⋅ S min V D − V A (5)

The FSR of the anisotropic MZI is designed to be 5 nm. At a bias voltage of 1.25 V and a driving peak-to-peak voltage of 1.5 V, the modulator has good linearity. Considering the bandwidth limitation of the measurement setup, a low-pass filter with bandwidth of 65 GHz is utilized in the simulation of eye-diagram tests. In the simulation, the detector has a responsivity is 1A/W, dark current is 0A, and thermal noise is 1e⁻²² A/Hz^.05. Figure 10 shows the eye diagram of the modulator transmitting a 120 Gbaud PAM4 signal at 1550 nm. When transmitting signal with a balanced-arm MZI modulator, the bit error rate of the eye diagram is 1.58 × 10⁻¹⁵, and the extinction ratio is 4.63 dB, the overall insertion loss of the modulator is 5.74 dB. When transmitting signal with an unbalanced-arm MZI modulator, the bit error rate of the eye diagram is 1.66 × 10⁻¹⁵, and the extinction ratio is 4.38 dB, the overall insertion loss of the modulator is 5.71 dB. The noise and jitter of the eye diagram in Figure 10 are very small, and the PAM4 eye diagram also has good linearity, and its RLM>0.922. In summary, circuit-level simulations show that the modulator functions well both statically and dynamically.

Table 4 shows the comparison between the state-of-the-art electro-optic modulators based on silicon nitride waveguide platform. As can be seen, our device exhibits the advantage of high bandwidth and low insertion loss. The simulated bandwidth is 174 GHz, which is much higher than that seen in the literature. The insertion loss of our device is 5.74 dB, which include two grating coupling loss of 3.08 dB. Compared with other polymer/silicon nitride hybrid designs [41, 28, 43], our design shows the advantages of high bandwidth, high modulation efficiency and low insertion loss. The length of the active region of the structure is only 3 mm. Although the micro-ring modulator is smaller in size, it is more sensitive to the manufacturing tolerance and temperature change. In addition to the deposition of organic polymers on silicon nitride, there are also the deposition of PZT or ZnO structures. But the annealing temperature required for PZT [25, 42] deposition is about 620°, and such a high temperature can lead to compatibility problems with the metal process. When ZnO [26] material is deposited by atomic layer technology, the electro-optical coefficient of the electro-optical material is small, and the phase shift efficiency of the device is poor. In addition, the bandwidth of the device is limited by the photon lifetime of the microring resonator and the RC time constant, and the bandwidth is very small, less than GHz. Although organic polymer-based modulators show good performance, organic polymer materials are not compatible with the current CMOS process, so the integration process of organic polymer materials with silicon nitride needs to be carefully designed and discussed.