  Recently, we have proposed the half-Josephson laser (HJL): a device that
combines lasing with superconducting leads, providing a locking between the
optical phase and the superconducting phase difference between the leads. In
this work, we propose and investigate two setups derived from a superconducting
quantum interference device (SQUID), where two conventional Josephson junctions
are replaced by two HJLs. In the first setup, the HJLs share the same resonant
mode, while in the second setup two separate resonant modes of the two lasers
are coupled optically. We dub the setup `light-superconducting interference
device' (LSID).
  In both setups, we find the operating regimes similar to those of a single
HJL. Importantly, the steady lasing field is significantly affected by the
magnetic flux penetrating the SQUID loop, with respect to both amplitude and
phase. This provides opportunities to tune or even quench the lasing by varying
a small magnetic field.
  For the second setup, we find a parameter range where the evolution equation
for the laser fields supports periodic cycles. The fields are thus modulated
with the frequency of the cycle resulting in an emission spectrum consisting of
a set of discrete modes. From this spectrum, two modes dominate in the limit of
strong optical coupling. Therefore, the LSID can be also used to generate such
modulated light.
