By studying high-precision ion yield measurements in diatomic molecules N-2 and NO with 30-fs linearly (LP) and circularly (CP) polarized light, we extend to molecular systems the study of ellipticity effects on both single and double ionization in strong laser fields. Competing ionization rates between LP and CP light, as predicted in the Ammosov-Delone-Krainov model, is observed experimentally by studying singly ionized N-2 and NO. We further observe an anomalously high NO2+ signal with CP, which shows a nonsequential ionization signature. A hypothesis is given that ties this observation to the influences of electronic structure on ellipticity effects in strong-field ionization.
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