Graphene has generated exceptional interest as an optoelectronic material 1,2 because its high carrier mobility 3,4 and broadband absorption 5 promise to make extremely fast and broadband electro-optic devices possible [6][7][8][9] . Electro-optic graphene modulators previously reported, however, have been limited in bandwidth to a few gigahertz 10-15 because of the large capacitance required to achieve reasonable voltage swings. Here, we demonstrate a graphene electro-optic modulator based on resonator loss modulation at critical coupling 16 that shows drastically increased speed and efficiency. Our device operates with a 30 GHz bandwidth and with a stateof-the-art modulation efficiency of 15 dB per 10 V. We also show the first high-speed large-signal operation in a graphene modulator, paving the way for fast digital communications using this platform. The modulator uniquely uses silicon nitride waveguides, an otherwise completely passive material platform, with promising applications for ultra-low-loss broadband structures and nonlinear optics.Integrated graphene modulators, by the nature of their electroabsorptive structure, carry fundamental tradeoffs between speed and efficiency. In these structures, graphene forms at least one electrode of a large capacitor. A voltage applied to this capacitor causes carriers to accumulate on the graphene sheet and gates the interband absorption of the graphene through Pauli blocking 17 . This change in absorption modulates the intensity of light travelling through the waveguide. Operation speed can be increased by using a thicker gate oxide, but the lower capacitance makes for a lower carrier concentration change with voltage and reduced efficiency.We overcome this tradeoff by exploiting critical coupling effects, whereby an increase in loss in a coupled resonator increases the system transmission by changing the condition for resonator coupling ( Fig. 1a). We have designed a resonator to be critically coupled for low losses. When losses are increased, the resonator becomes undercoupled, increasing transmission through the bus waveguide. This effect has been used to create sensitive all-optical switches 16,18 . Here, above a segment of a silicon nitride ring resonator, we integrate a graphene/graphene capacitor to modulate the round-trip ring loss (Fig. 1b). At 0 V bias, both graphene sheets in the capacitor are lightly doped and thus opaque, so the ring has high loss and is undercoupled to the bus waveguide. Applying a voltage to the capacitor dopes the graphene sheets heavily, causing their absorption to decrease as the Fermi level crosses half the incident photon energy. The ring-now with substantially lower losscouples to the bus waveguide, decreasing the system's transmission, as predicted theoretically 19 . Sensitivity to ring loss and on-state insertion loss can be designed by choosing the ring-waveguide coupling constant ( Supplementary Figs 1 and 2). We note that this mechanism is not simply ring-enhanced absorption modulation, as the ring has little circulating p...

The ability to render objects invisible using a cloak - not detectable by an external observer - for concealing objects has been a tantalizing goal1-6. Here, we demonstrate a cloak operating in the near infrared at a wavelength of 1550 nm. The cloak conceals a deformation on a flat reflecting surface, under which an object can be hidden. The device has an area of 225 um2 and hides a region of 1.6 um2. It is composed of nanometre size silicon structures with spatially varying densities across the cloak. The density variation is defined using transformation optics to define the effective index distribution of the cloak.Comment: 12 pages, 5 figure

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