We investigate the microwave characteristics of a spin transfer nano-oscillator (STNO) based on coupled vortices as a function of the perpendicular magnetic field $H_perp$. While the generation linewidth displays strong variations on $H_perp$ (from 40 kHz to 1 MHz), the frequency tunability in current remains almost constant (~7 MHz/mA). We demonstrate that our vortex-based oscillator is quasi-isochronous independently of $H_perp$, so that the severe nonlinear broadening usually observed in STNOs does not exist. Interestingly, this does not imply a loss of frequency tunability, which is here governed by the current induced Oersted field. Nevertheless this is not sufficient to achieve the highest spectral purity in the full range of $H_perp$ either: we show that the observed linewidth broadenings are due to the excited mode interacting with a lower energy overdamped mode, which occurs at the successive crossings between harmonics of these two modes. These findings open new possibilities for the design of STNOs and the optimization of their performance.
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