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 $\lambda =1.55\mu \mathrm{m}$ 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 $460\mathrm{nm}$ . The spectral response of the detector extends well beyond $1.6\mu \mathrm{m}$ at $300\mathrm{K}$ . The absorption length of $\sim 135\mu \mathrm{m}$ (at 1/e decrease of intensity) at $\lambda =1.55\mu \mathrm{m}$ along the waveguide allows for relatively small-size devices for all-on-one-platform integration.