In this paper, we present results on a master-oscillator Yb-doped fiber amplifier with 1 kW cw output power (at 1064nm), and near-diffraction limited beam quality (M2<1.4), with internal quantum efficiency >83%. The final amplifier stage uses a very high Yb-doped 35-um core LMA fiber, using a new process recipe that virtually eliminates photo-darkening. As a result, high efficiency, SBS-free operation to 1 kW cw power level is obtained, with a phase modulation bandwidth of only 450MHz, well below other reported results. To enable single-frequency cw power scaling to kW levels, we investigate LMA fiber waveguide designs exploiting gain-discrimination, using partially Yb-doped LMA fiber cores, with various diameters up to 80-um. SBS-free, singlefrequency (few kHz) operation is demonstrated up to 0.9kW cw power. At the lower cw powers (<200W) neardiffraction limited beam-quality is obtained, but is observed to deteriorate at higher cw powers. We discuss potential causes, and present a detailed simulation model of kW large-core fiber-amplifiers, that includes all guided modes, fiber bend, transverse spatial hole burning, gain-tailoring, mode-scattering, SBS nonlinearity, and various thermal effects. This model shows good agreement with the observed single-frequency power scaling and beam-quality characteristics.© (2011) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.