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This work demonstrates a rib waveguide photodetector based on a vertical Si p-i-n junction with Ge islands operating in the spectral region around lambda=1.55 microm at room temperature. A vertical stack of four layers of Ge islands is grown by molecular beam epitaxy on a silicon-on-insulator. Each layer is organized in a two-dimensional square grid with a period of 460 nm. The spectral response of the detector extends well beyond 1.6 mum at 300 K. The absorption length of approximately 135 microm (at 1/e decrease of intensity) at lambda=1.55 microm along the waveguide allows for relatively small-size devices for all-on-one-platform integration.