High-order harmonics generated by bicircular laser field have helicities which alternate between $+1$ and $-1$. In order to generate circularly polarized high-harmonic pulses, which are important for applications, it is necessary to achieve asymmetry in emission of harmonics having opposite helicities. We theoretically investigated a wide range of bicircular field component intensities and found areas where both the harmonic intensity is high and the helicity asymmetry is large. We investigated the cases of $omega$--$2omega$ and $omega$--$3omega$ bicircular fields and atoms having the $s$ and $p$ ground states, exemplified by He and Ne atoms, respectively. We have shown that for He atoms strong high harmonics having positive helicity can be generated using $omega$--$3omega$ bicircular field with a much stronger second field component. For Ne atoms the helicity asymmetry can be large in a wider range of the driving field component intensities and for higher harmonic orders. For the stronger second field component the harmonic intensity is higher and the helicity asymmetry parameter is larger for higher harmonic orders. The results for Ne atoms are illustrated with the parametric plots of elliptically polarized attosecond high-harmonic field.
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