By embedding graphene sheet in the silicon waveguide, the overall effective mode index displays unexpected symmetry and the electrorefraction effect has been significantly enhanced near the epsilon-near-zero point. An eight-layer graphene embedded Mach-Zehnder Modulator has been theoretically demonstrated with the advantage of ultracompact footprint (4 × 2？μm2), high modulation efficiency (1.316？V·μm), ultrafast modulation speed, and large extinction ratio. Our results may promote various on-chip active components, boosting the utilization of graphene in optical applications. 1. Introduction Electrooptical modulator, which transfers electronic signals into high bit-rate photonic data, is the key component in on-chip interconnection and integrated optoelectronic circuits [1]. By applying an electric field to a material, the real and imaginary part of the refractive indices can be changed. A change in the real part of the refractive index caused by the applied voltage is known as the electrorefraction (ER) effect, whereas a change in the imaginary part of the refractive index is known as the electroabsorption (EA) effect [1]. However, these two effects are too weak in pure silicon at the communication wavelengths so that it usually needs an extremely large length to reach the required modulation; for example, a 50？Gbit/s modulator has the length of 1 millimeter [2]. The large footprint of optical modulator makes it impossible to be integrated into a single chip. To fill the demands of next generation on-chip communication, minimizing the size and improving the speed of modulator become the urgent goals but remain a challenge. Recently, graphene-based modulators have attracted much attention due to their unprecedented ability to enhance the material’s EA effect; thus, they can greatly reduce the modulator length to achieve the same effect [2–5]. In [5], Lu and Zhao reported a graphene embedded modulator having a length of only 1？μm. However, limited by the inbuilt drawback from the EA modulator, the extinction ratio is low and background noise cannot be ignored. In this paper, we point out that graphene can also have significant enhancement to the ER effect of the background material. By embedding duplicated graphene layers into a silicon substrate, we demonstrated that the variation of effective mode index can be as large as 0.4057 corresponding to a Mach-Zender modulation arm length of only 1.9？μm. Note that caused by ER effect is only at the level of 10？4 for conventional semiconductor modulators [6–8]. To our knowledge, this is the largest and smallest
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