Misalignment effects in nonlinear feedback devices

In this paper a broad class of systems whose feedback loop is misaligned. This is the case, for instance, when the external mirror is not perfectly orthogonal to the beam axis, so that the transverse profile of the re-injected beam is laterally shifted. The optical field in a point E(x) within the device is then coupled with E_Deltax = E(x + Deltax), with Deltax lateral shift. This nonlocal coupling is the spatial analogous of temporal delay where the field at time t is coupled with the delayed field at t + Deltat. In this work we focus on spatial effects, assuming that the in the feedback loop is negligible as compared with temporal scales of the dynamics of the nonlinear device.