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We design a double-clad fiber with a W-type core for use in a cladding-pumped fiber Raman amplifier operating at the 1st-Stokes wavelength. Compared to a conventional step-index core, a W-type core allows for a larger inner cladding so that pump sources with relatively poor beam quality can be utilized. This is because parasitic Raman conversion to the 2nd Stokes can be suppressed by bending loss, while the sharp spectral characteristics of a W-type fiber allows the loss to be kept low at the 1st Stokes. Without bendloss filtering, the ratio between the inner-cladding area and the effective area of the Stokes wave is limited to approximately eight. With a bendloss-filtered W-type fiber, area ratios of 34 are possible for core diameters up to around 18 μm and inner-cladding diameters up to around 82 μm. We also discuss limitations on the brightness enhancement due to optical damage and pump-signal walk-off in the pulsed regime.