Recovering pyramid WS gain in non-common path aberration correction mode via deformable lens

It is by now well known that pyramid based wavefront sensors, once in closed loop, have the capability to improve more and more the gain as the reference natural star image size is getting smaller on the pyramid pin. Especially in extreme adaptive optics applications, in order to correct the non-common path aberrations between the scientific and sensing channel, it is common use to inject a certain amount of offset wavefront deformation into the DM(s), departing at the same time the pyramid from the optimal working condition. In this paper we elaborate on the possibility to correct the low order non-common path aberrations at the pyramid wavefront sensor level by means of an adaptive refractive lens placed on the optical path before the pyramid itself, allowing the mitigation of the gain loss.