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. 2020 Jun 17;11:3073. doi: 10.1038/s41467-020-16812-6

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 2 — a Electromechanical S₁₁ spectrum from 1 to 5.5 GHz. A range of equidistant bulk acoustic resonances is found to exist over a broad frequency range. The inset schematic illustrates the acoustic wave reflection at interfaces. b, c Optomechanical S₂₁ responses of TE and TM modes demonstrate acousto-optic modulation covering multiple octave-spanning microwave frequencies. Due to different optical mode shapes (insets in b and c) and thus acousto-optic mode overlap, TE and TM modes show dissimilar S₂₁ spectra. d, e The zoom-in of S₁₁ and TE mode’s S₂₁ within the window (green shaded area in b) around 2 GHz. f The zoom-in of TM mode’s S₂₁ around 4 GHz in c. The resonances distribute evenly with an FSR of 17.5 MHz. g Schematic of the setup for measuring electromechanical and optomechanical response. DUT device under test, ECDL external cavity diode laser, FPC fiber polarization controller, PD photo-diode, VNA vector network analyzer. h Numerical simulation of vertical stress σ_z distribution for one typical acoustic resonant mode at 2.041 GHz under 1 V driving field, with a zoom-in around the optical waveguide (red box in h) shown in i. At several GHz, the acoustic wavelength is similar in scale to optical wavelength and waveguide structure. j The measured (circle) and calculated (solid line) frequency difference between each pair of adjacent Si resonances, showing a periodic variation of local FSR around an average value of 17.5 MHz. The maxima of FSR align with the SiO₂ resonances marked by green dashed lines. k Measured (circle) and calculated (solid line) higher order dispersion represented by the frequency deviation from an equidistant frequency grid (with average FSR = 17.515 MHz), referencing to mode ν₀ (3.0145 GHz). μ is the mode number difference relative to the mode at 3.0145 GHz.