Optical modulators are essential building blocks for high-capacity optical communication and massively parallel computing. Among all types of optical modulators, traveling-wave Mach-Zehnder modulators (TW-MZMs) featuring high speed and efficiency are widely used, and have been developed on a variety of integrated material platforms. Existing methods to design and simulate TW-MZMs so far strongly rely on specific material properties, and thus inevitably involve complicated electrical-circuit models. As a result, these methods diverge significantly. In addition, they become increasingly inefficient and inaccurate for TW-MZMs with extended length and rising modulation speed, posing formidable challenges for millimeter-wave and terahertz operation. Here, we present an innovative perspective to understand and analyze high-speed TW-MZMs. Our perspective leverages nonlinear optics and complex band structures of periodic rf electrodes, and is thus entirely electromagnetic-wave-based. Under this perspective, we showcase the design, optoelectronic simulation, and experimental validation of high-speed TW-MZMs based on $\mathrm{Si}$ and ${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$, and further demonstrate unambiguous advantages in simplicity, accuracy, and efficiency over conventional methods. Our approach can essentially be applied to almost any integrated material platform, including those based on semiconductors and electroabsorption materials. With high-frequency electrode designs and optoelectronic co-simulation, our approach facilitates the synergy and convergence of electronics and photonics, and offers a viable route to construct future high-speed millimeter-wave and terahertz photonics and quantum systems.

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• Electro-optic effects
• Integrated optics
• Nonlinear optics
• Photonic crystals
• Devices
• Optical techniques