  We investigate the effect of magnetic field on a photonic-crystal Josephson
traveling-wave parametric amplifier (TWPA). We show that the observed change in
photonic bandgap and plasma frequency of the TWPA can be modeled by considering
the suppression of the critical current in the Josephson junctions (JJs) of the
TWPA due to the Fraunhofer effect and closing of the superconducting gap.
Accounting for the JJ geometry is crucial for understanding the field
dependence. In one in-plane direction, the TWPA bandgap can be shifted by 2 GHz
using up to 60 mT of field, without losing gain or bandwidth, showing that
TWPAs without SQUIDs can be field tunable. In the other in-plane direction, the
magnetic field is perpendicular to the larger side of the Josephson junctions,
so the Fraunhofer effect has a smaller period. This larger side of the JJs is
modulated to create the bandgap. The field interacts more strongly with the
larger junctions, and as a result, the TWPA bandgap closes and reopens as the
field increases, causing the TWPA to become severely compromised already at 2
mT. A slightly higher operating limit of 5 mT is found in out-of-plane field,
for which the TWPA's response is hysteretic. These measurements reveal the
requirements for magnetic shielding needed to use TWPAs in experiments where
high fields at the sample are required; we show that with magnetic shields we
can operate the TWPA while applying over 2 T to the sample.
