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In this paper, we analyze the optical conversion efficiency of a multimode pump source into a diffraction-limited beam in cladding-pumped Raman fiber lasers and amplifiers. We theoretically derive a relation of the cladding/core area ratio for efficient energy transfer from the pump into the first Stokes while avoiding the second Stokes generation that impedes the conversion efficiency of the first Stokes. In addition, we investigate experimentally the limits on the conversion efficiency in a pulse-pumped double-clad Raman fiber. We obtained a peak power conversion efficiency into the first Stokes in excess of 75% and a pulse energy conversion of 60%. We found that spatial pump mode depletion is the main limiting factor in this fiber, with concentric cladding and core. We discuss various solutions to overcome these limitations and improve on the current results.