A quasi-matching scheme for arbitrary group velocity match in electro-optic modulation

Ultrafast electro-optic (EO) modulators are one of the building blocks of integrated photonics, playing a key role in data communication,^1–3 microwave photonics,⁴ and quantum technologies.⁵ In recent years, various high-speed EO modulators have been developed, including Mach–Zehnder modulators^6,7 (MZM), micro-ring modulators,⁸ Fabry–Perot cavity modulators,⁹ and kappa modulators.¹⁰ Among them, MZM has the advantages of low driving voltage, high extinction ratio, large bandwidth, large manufacturing tolerance, and good stability, making it a favorable choice in integrated systems.^1,2,11

MZMs usually have a relatively long modulation length so traveling wave electrodes (TWEs) are required.^6,12 Under this configuration, group velocity match between microwave and light in a MZM should be carefully designed to guarantee the maximum of EO interaction. In addition, the characteristic impedance of TWEs should be optimized to match radio frequency (RF) cables and equipment to reduce the undesired reflection. Various approaches have been employed to meet the two requirements. On a lithium niobate on insulator (LNOI) platform, a co-planar waveguide (CPW) electrode was developed to realize both the velocity and impedance match condition through carefully optimizing structural parameters when the substrate is silicon.^1,2 On a III–V semiconductor platform, the group velocity of the light wave is slower than the RF wave in a conventional CPW electrode, hence the capacitor-loaded electrode (CLE) is introduced to slow down the RF group velocity while maintaining characteristic impedance equal to 50 Ω.¹³ On a LN on the quartz substrate platform, an RF wave travels faster than optical mode due to the low permittivity of the quartz substrate, so the CLE is preferred.¹⁴ 

However, velocity and impedance matches are not necessary to be realized at some platforms.¹⁵ For example, both the conventional CPW electrode with a fast-traveling speed and CLE with a slow-traveling speed cannot achieve group velocity matches under the circumstance of impedance match on the popular LNOI platform with a silica cladding.¹⁵ The background reason is a group refractive index gap existing between fast-wave and slow-wave TWEs. Partially removing silicon substrate,^15,16 making compensation at optical domain,¹⁷ and realizing group velocity matching at the cost of impedance mismatch¹⁸ have been utilized to improve the modulation performance; however, none of the schemes can be served as a universal solution. In this paper, we propose a general method of quasi-velocity-matching scheme by cascading fast-wave and slow-wave TWEs on a LNOI platform with a silica cladding, where the optical group velocity just falls in the group velocity gap between the slow-wave and fast-wave TWEs. The effective group velocity of the quasi-matching scheme can be flexibly adjusted by changing the ratio of fast-wave and slow-wave TWEs, and the RF insertion loss at the boundary of the slow-wave and fast-wave TWEs is less than 0.12 dB. Moreover, the quasi-matching scheme is experimentally verified by demonstrating 6 mm long EO modulators. The measured small signal EO response of the modulator with a cascaded TWE drops less than 2 dB at 67 GHz, while the measured small-signal EO response of modulators with conventional slow-wave and fast-wave TWEs drop 4 dB at 67 GHz. As a result, we anticipate that our proposed quasi-matching scheme will serve as a promising method for the design of MZMs in different kinds of integration platforms.

We designed the EO modulator with the cascaded TWE on an X-cut 400 nm LN on a 3 µm buried oxide platform with a silica cladding as shown in Fig. 1(d) to verify the proposed group velocity matching method. This kind of platform with electrodes deposited on the cladding is highly desired in many applications because the topology of electrodes can be flexibly designed without worrying the crossing with waveguides. Under the consideration of real fabrication, the waveguide configuration is set as height h_rib = 200 nm, slab thickness h_slab = 200 nm, angle θ = 60°, and width w = 900 nm to ensure single-mode operation, as shown in Fig. 1(d). The optical group refractive index of such a waveguide is n_g = 2.22. To minimize the metal absorption loss, the minimum gap between metals and the thickness of silica cladding are chosen as 3.1 µm and h_clad = 650 nm, respectively. The simulated metal absorption loss is less than 0.01 dB/cm under these conditions.

The RF loss of TWEs is indeed an important issue in the design of EO modulators. When using a cascaded TWE, reflection and insertion loss at the junction of slow-wave and fast-wave TWEs need to be taken into consideration seriously. Reflection and insertion loss are mainly resulted from characteristic impedance and modal mismatches between the two adjoint TWEs. Therefore, both the slow-wave and fast-wave TWEs should be designed with characteristic impedances of 50 Ω. Given that the current in the slow-wave TWE is constricted within the signal plate, the current around the slow-wave TWE hardly endures distortion when entering the fast-wave TWE, so the slow-wave and the fast-wave TWEs with the same signal plate widths, denoted as w_cpw = w_cle, can reduce the modal mismatch.

The finite difference element method is adopted to design the electrical properties of the proposed TWEs. Confined by the minimum exposure linewidth of ultraviolet lithography and the lift-off process, the minimum size of the TWE is set to 2 µm and the thickness of Au is set to d = 600 nm. In order to ensure that the RF loss is within an acceptable range, the width of the signal plate is set to be larger than 10 µm. Under the condition of the characteristic impedance of 50 Ω and g_cle = 3.1 µm, the slow-wave electrode is designed with an RF group refractive index as close to n_g as possible. After careful optimization, the slow-wave electrode parameters are chosen as (R, c, t, s, l, w_cle) = (45, 5, 5, 3, 3, 18 µm). As for fast-wave TWE, we get g_cpw = 4.5 µm to realize the characteristic impedance of 50 Ω according to w_cpw = w_cle.

Figure 2(a) shows the calculated propagation loss as a function of the frequency of slow-wave and fast-wave TWEs, where the slow-wave TWE has a smaller propagation loss as expected. Figure 2(b) shows the simulated impedance as a function of the frequency of slow-wave and fast-wave TWEs, indicating the impedance matching over a broadband frequency range. Figure 2(c) shows the simulated group refractive index as a function of the frequency of slow-wave, fast-wave, and cascaded TWEs, clearly indicating that the effective group velocity of cascaded TWE falls in the gap between slow-wave and fast-wave TWEs. The value of effective group velocity can be tuned by the ratio of fast and slow TWEs. Finally, the modulation efficiency of EO modulators with fast-wave, slow-wave, and cascaded TWEs are calculated as V_π · L = 3.03 V cm, V_π · L = 2.33 V cm, and V_π · L = 2.64 V cm, respectively.

The EO modulation bandwidths of slow-wave, fast-wave, and cascaded TWEs are analyzed. The cascading period of the cascading TWE is fixed as 3 mm. Figures 2(d)–2(f) shows the EO modulation response of the fast-wave (red), slow-wave(blue), and cascading (black) TWEs for EO modulators with modulation lengths of 6 mm in Fig. 2(d), 9 mm in Fig. 2(e), and 15 mm in Fig. 2(f), respectively. As shown in Fig. 2(d), the 3-dB modulation bandwidth of the EO modulator with a slow-wave or fast-wave TWE is less than 70 GHz because of the mismatch of group velocity, but the cascaded TWE scheme exhibits a 3-dB modulation bandwidth of over 100 GHz when the group velocity is quasi-matched. Moreover, this improvement can be observed for all three modulation lengths mentioned. In addition, we calculate the EO response of the cascaded modulator with a different cascaded number as shown in Fig. 2(g). The result shows that the larger the cascaded number, the better the EO bandwidth. In order to investigate how the discrepancy between n_cpw and n_cle affect the modulation bandwidth, we use δ to define the discrepancy of the RF group index between the fast-wave and slow-wave TWEs. The group velocity of n_cpw and n_cle are replaced by n_cpw′ = n_cpw − δ and n_cle′ = n_cle + δ while keeping other parameters unchanged. Under these configurations, although the overall index is still 2.22, the modulator with smaller RF group index discrepancy exhibits better the modulation performance, as shown in Fig. 2(h).

Experimentally, we fabricated the fast-wave, slow-wave, and cascaded TWEs on an X-cut 400 nm LN on a 3 µm buried oxide platform with a silicon substrate. The waveguide is defined by electron beam lithography (EBL), followed by the ion coupled plasmonic (ICP) etching process. After that, 650 nm thickness of silica is deposited by plasma enhanced chemical vapor deposition (PECVD). To reduce the RF attenuation caused by the PECVD-deposited silica, the sample is annealed under 700 °C in vacuum for 3 h. Finally, the electrode is made using a lift-off process.

The static EO signal is measured using a 1-MHz triangular voltage sweep method. Figures 3(a)–3(c) show the normalized optical transmission of the fast-wave, slow-wave, and cascaded TWE modulators. The modulation efficiency for the fast-wave, slow-wave, and cascaded TWE modulators are calculated as V_π · L = 3.46 V cm, V_π · L = 3.94 V cm, and V_π · L = 3.65 V cm, respectively. The modulation efficiency of the cascaded TWE modulator is the superposition of the fast- and the slow-wave TWE modulators. The discrepancy between the measured V_π · L and the simulated V_π · L is introduced by the annealing process to purify the PECVD silica. Such discrepancy can be eliminated by improving the fabrication process.

The microwave properties of three types of TWEs were measured by a pair of 67-GHz RF probes connecting TWEs and RF cables. The group RF refractive indices of the fast-wave (red), slow-wave (blue), and cascaded (green) TWEs are shown in Fig. 4(a). The measured RF refractive index of the cascaded TWE falls in the gap between fast-wave and slow-wave TWEs and is consistent with the predicted value (dashed) under the ratio k of 0.438. This finding confirms that the effective refractive index can be altered by adjusting the length of the two types of TWEs when cascading the fast-wave and slow-wave TWEs. Figure 4(b) shows the characteristic impedance of the fast-wave (red), slow-wave (blue), and cascaded (black) TWEs, conveying the information that the fabricated TWEs are well satisfied with the characteristic impedance match condition. Figure 4(c) shows the measured propagation loss as a function of the frequency of fast-wave (red), slow-wave (blue), and cascaded (black) TWEs, indicating similar propagation losses of all kinds of TWEs in a broad range of frequency. The reason for the discrepancy with simulation results where different kinds of TWEs have different propagation losses is that the experimental propagation losses are mainly dominated by the absorption of PECVD silica. In addition, we also paid close attention to the insertion loss at the junction of slow-wave and fast-wave TWEs. We fabricated cascaded TWEs with the same total length of slow-wave and fast-wave TEWs but a different number of junctions, as shown in the inset of Fig. 4(d); the insertion loss at the juncture is less than 0.12 dB by comparing the measured results with the different number of junctions.

In order to verify the advantage of cascaded TWE in EO modulation, we fabricated 6 mm EO modulators with fast-wave, slow-wave, and cascaded TWEs. Figure 5(a) shows the experimental setup to measure the EO modulation bandwidth. A 1550 nm laser with polarization adjusted by a polarization controller is coupled into the waveguide through the grating coupler after passing by a variable optical attenuator and an erbium-doped fiber amplifier. The RF signal was applied to the TWEs through a 67 GHz RF probe and the RF signal after transmitting to the device is terminated by attaching a second RF probe connected with a 50 Ω terminator. The modulated light was analyzed by a 67 GHz light component analyzer and the measured signals were shown in Fig. 5(b). The fast-wave and slow-wave modulators exhibit a modulation bandwidth of around 50 GHz, which is constricted by the mismatch of group velocity. On the contrary, the EO response of the cascaded modulator drops less than 2 dB at 67 GHz. These results are consistent with the simulations, demonstrating the advantage of the quasi-matching scheme over the unmatched scheme.

Finally, we measured the eye diagrams for the 100 Gbps on-off-key (OOK) signals of modulators with slow-wave and cascaded TWEs. Figure 6(a) shows the setup for measuring the modulator’s eye diagrams. The electrical signal generated by a clock source was transformed into the OOK signal by an arbitrary waveform generator. The OOK signal combined with static voltage using a bias-tee was then applied to the fabricated device, and the modulated optical signal was detected by a photodetector and subsequently connected to a real-time oscilloscope. As shown in Figs. 6(b)–6(d), the openness of the eye diagram of the cascaded modulator is larger than the fast-wave and slow-wave modulator, showing that the cascaded modulator has a better performance under the same measurement condition.

In conclusion, we have proposed and demonstrated a quasi-matching scheme by cascading fast-wave and slow-wave TWEs for arbitrary group velocity while maintaining impedance match. The new TWE scheme exhibits a quasi-matched effective RF refractive index and a neglectable insertion loss at the junction of fast and slow wave TWEs. The advantages of the cascaded TWE in EO modulation were verified by comparing the EO modulation bandwidth and speed of EO modulators with slow-wave, fast-wave, and cascaded TWEs on a LNOI platform with silica cladding. Therefore, such a group velocity matching method is expected to boost the emergence of high-speed modulators in different kinds of material platforms, finding many applications in optical communications and microwave photonics.

This work was supported by the National Key Research and Development Program of China (Grant No. 2022YFB2803800), the National Natural Science Foundation of China (Grant Nos. 62105283 and 62205286), the Zhejiang Provincial Natural Science Foundation of China (Grant No. LDT23F04012F05), and the Leading Innovative and Entrepreneur Team Introduction Program of Zhejiang (Grant No. 2021R01001).

The authors have no conflicts to disclose.

Siyuan Wang: Conceptualization (equal); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Hongxuan Liu: Investigation (supporting); Methodology (supporting); Validation (supporting). Mai Wang: Investigation (supporting); Methodology (supporting). Hao Chen: Investigation (supporting). Zhi Ma: Investigation (supporting). Bingcheng Pan: Investigation (supporting). Yishu Huang: Investigation (supporting). Yaqi Shi: Investigation (supporting). Chenlei Li: Investigation (supporting). He Gao: Investigation (supporting). Yeyu Tong: Investigation (supporting); Resources (supporting). Zongyin Yang: Resources (supporting). Zejie Yu: Conceptualization (equal); Supervision (lead); Writing – review & editing (equal). Liu Liu: Methodology (supporting); Supervision (supporting). Daoxin Dai: Supervision (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.