  Nonlinear interferometers are promising tools for quantum metrology, as they
are characterized by an improved phase sensitivity scaling compared to linear
interferometers operating with classical light. However, the multimodeness of
the light generated in these interferometers results in the destruction of
their phase sensitivity, requiring advanced interferometric configurations for
multimode light. Moreover, in contrast to the single-mode case, time-ordering
effects play an important role for the high-gain regime in the multimode
scenario and must be taken into account for a correct estimation of the phase
sensitivity. In this work, we present a theoretical description of spatially
multimode SU(1,1) interferometers operating at low and high parametric gains.
Our approach is based on a step-by-step solution of a system of
integro-differential equations for each nonlinear interaction region. We focus
on interferometers with diffraction compensation, where focusing elements such
as a parabolic mirror are used to compensate for the divergence of the light.
We investigate plane-wave and Gaussian pumping and show that for any parametric
gain, there exists a region of phases for which the phase sensitivity surpasses
the standard shot-noise scaling and discuss the regimes where it approaches the
Heisenberg scale. Finally, we arrive at insightful analytical expressions for
the phase sensitivity that are valid for both low and high parametric gain and
demonstrate how it depends on the number of spatial modes of the system.
