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Core-pumped silicon Raman amplifiers reported so far have exhibited gains of 4 dB or less in continuous-wave operation, limited by free-carrier absorption. Here we propose a device design that mitigates the problem of free-carrier absorption and leads to significantly higher gain values. In the new design, we surround the silicon waveguide with an additional rib-shaped passive cladding with a refractive index between that of silicon and the silica buffer. By injecting the pump power into the cladding, instead of directly into the silicon core, the overlap of the pump power with the silicon core is reduced and thus the effect of free-carrier absorption on the pump is weakened. The amplifier may therefore be made much longer, and higher total gains be achieved than in conventional core-pumped amplifiers. We present a detailed analysis of a design where a total gain as high as 18 and 34 dB may be achieved with pump powers of 300 mW and 1 W, respectively. We describe the model we have developed to design and optimize cladding-pumped silicon Raman amplifiers. Explicit expressions are formulated for the nonlinear effective areas for stimulated Raman scattering and two-photon absorption in silicon waveguides, taking into account the differing tensorial structures of these nonlinearities.