  High harmonic generation (HHG) is a unique and useful process in which
infrared or visible radiation is frequency up converted into the extreme
ultraviolet and x ray spectral regions. As a parametric process, high harmonic
generation should conserve the radiation energy, momentum and angular momentum.
Indeed, conservation of energy and momentum have been demonstrated. Angular
momentum of optical beams can be divided into two components: orbital and spin
(polarization). Orbital angular momentum is assumed to be conserved and
recently observed deviations were attributed to propagation effects. On the
other hand, conservation of spin angular momentum has thus far never been
studied, neither experimentally nor theoretically. Here, we present the first
study on the role of spin angular momentum in extreme nonlinear optics by
experimentally generating high harmonics of bi chromatic elliptically polarized
pump beams that interact with isotropic media. While observing that the
selection rules qualitatively correspond to spin conservation, we unequivocally
find that the process of converting pump photons into a single high-energy
photon does not conserve angular momentum, i.e. this process is not self
contained. In one regime, we numerically find that this major fundamental
discrepancy can be explained if the harmonic photons are emitted in pairs. Yet
in another regime, the ionizing electron carries the missing angular momentum.
The results presented here, apart from exploring the very foundations of HHG,
are also important for a variety of applications, as our system exhibits full
control over the harmonics polarization, from circular through elliptical to
linear polarization, without comprising the efficiency of the process. This
work paves the way for a broad range of applications with HHG, from ultrafast
circular dichroism to zepto-clocks and to attosecond quantum optics.
