Due to the current explosive growth of data traffic, optical transmission systems need to move towards the direction of high speed and low power consumption through modulation. Miniaturized electro-optic modulators are the key components of modern electrical communication networks and microwave-photon systems, which can convert electrical signals into optical domains. Traditional monolithic integrated electro-optic modulators require long electrodes to induce large optical phase shifts and therefore require a trade-off between electro-optic bandwidth and half-wave voltage, which cannot be met with large bandwidth and low voltage simultaneously. To address the above problem, researchers have proposed various solutions with different materials and waveguide structures. However, existing structures are generally weakly bound to the electric fields, and the mutual coupling strength between optical wave and microwave is limited, resulting in low modulation efficiency and large device size. If the modulated microwave can be enhanced to increase the coupling strength between electromagnetic fields in the waveguide, the optical wave can obtain a larger phase shift when the same driving voltage is applied. Therefore, we wish to propose a structure that can dramatically increase the electro-optic overlap volume and coupling strength between optical wave and microwave, so as to improve the electro-optic modulation efficiency and overcome the trade-off between bandwidth and voltage.

In this paper, a model based on the slow-light effect of spoof surface plasmonic polaritons is utilized to obtain a thin-film lithium niobate Mach-Zehnder interferometer electro-optic modulator. Taking advantage of the large electro-optic coefficient of lithium niobate and the characteristics of metal photonic crystal to limit the microwave within the sub-wavelength scale, the electro-optic modulator can significantly enhance efficiency. Moreover, we can flexibly adjust structural parameters to achieve the desired function according to our demands. Specifically, we realize the speed matching condition between optical waves and microwaves at the band edge of the metal photonic crystal by modulating the dispersion relation and group velocity of microwaves on the thin-film lithium niobate photonic chip. The eigenmode field distribution, group velocity, and dispersion relation of optical waves and microwaves are simulated and calculated by the eigenfrequency module of COMSOL Multiphysics software. The structure is then optimized to enhance the slow-light effect at the band edge of the metal photonic crystal. By enhancing the slow-light effect, the microwave mode field can be compressed in the sub-wavelength scale, leading to a significant improvement in the coupling intensity and overlap volume between the optical wave and microwave. This ultimately intensifies the linear electro-optic effect in the thin-film lithium niobate waveguide.

To demonstrate that the slow-light effect of a metal photonic crystal can improve the modulation efficiency, we calculate the electro-optic overlapping integration factor within lithium niobate waveguide in the three-dimensional space corresponding to structures with different parameters based on the relevant electric field of eigenmodes. Since microwave eigenmode is inhomogeneous in three dimensions, the electro-optic overlapping factor needs to be integrated into a volume. In addition, the modulation results are compared at the same modulated microwave frequency of 220 GHz. Specifically, the result shows that the structure featuring a stronger slow-light effect has a larger electro-optic overlapping integration factor (Fig. 3). Consequently, the proposed device with metal photonic crystal electrodes allows for a shorter propagation length to realize a more efficient modulation process when changing the same phase shift π in one arm of the Mach-Zehnder interferometer compared with the bar electrode structure. As a result, the scheme we proposed not only satisfies the need for miniaturization and integration but also reduces the total loss of the device with a shorter length, which can eventually realize an electro-optic modulator with large bandwidth and low driving voltage.

The slow-light effect of the spoof surface plasmonic polaritons structure provides a new idea and method to improve the modulation efficiency of thin-film lithium niobate electro-optic modulator chips. By modulating the structural parameters of the metal photonic crystal electrodes, the speed matching condition of optical waves and microwaves is satisfied. The group velocity of microwaves can be flexibly designed, and the electric field of microwaves at the band edge of dispersion can be enhanced and limited in sub-wavelength scale due to the characteristic of slow-light effects, which significantly increases the electro-optic overlap volume and mutual coupling strength. Subsequently, the slow-light effect structure allows a significant reduction in device length, which is of great importance for achieving high modulation efficiency and more compact photonic chips. The design concept of spoof surface plasmonic polaritons structure in this paper can be applied to not only thin-film lithium niobate photonic chips but also other material platforms with high electro-optic coefficients in the future. It also provides a new direction for the development of miniaturized integrated chips from the physical principles of optics and is expected to achieve more complex and diverse photonic circuits integrated with other functional devices.