▎ 摘　　要

All-optical graphene-based optical modulators have recently attracted much attention because of their ultrafast and broadband response characteristics (bandwidth larger than 100 GHz) in comparison with the previous graphene-based optical modulators, which are electrically tuned via the graphene Fermi level. Silicon photonics has some benefits such as low cost and high compatibility with CMOS design and manufacturing technology. On the other hand, graphene has a unique large nonlinear Kerr coefficient, which we calculate using graphene's tight-binding model based on the semiconductor Bloch equations. Its real and imaginary parts are negative at the wavelength of 1.55 mu m and E-F = 0.1 eV. To simultaneously use the benefits mentioned above, we present an all-optical, CMOS-compatible, and graphene-on-silicon slot (GOSS) waveguide extinction and phase modulator that consists of two different geometries. The first one consists of a one-stage GOSS waveguide with a single layer of graphene. To increase the light-graphene interaction and consequently enhance the modulation efficiency (ME), another stage of the GOSS waveguide is placed over the first one. This two-stage configuration is called a graphene-on-silicon double-slot (GOSDS) waveguide. The ME, insertion loss (IL), and modulation depth (MD) for a 12.5 mu m GOSDS waveguide modulator with a double layer of graphene can reach 0.241 dB/mu m, 1.31 dB, and 77%, respectively, at optical pump intensities about 9 MW cm(-2). Our design has a smaller waveguide length (17.6 times) than the previous all-optical graphene-on-silicon ribbon waveguide extinction modulator and high MD (about 2 times) in comparison with a graphene-dad microfiber all-optical extinction modulator. Compared with an all-optical Mach-Zehnder interferometer phase modulator, our design has short graphene coated waveguide length (approximate to 0.1 times) and low local optical intensities (approximate to 0.043 times) needed for pi phase shift. This study may promote the design and realization of high-performance, wideband, compact, and all-optical control on a single chip with a reasonable contrast level. (C) 2021 Optical Society of America