  Context. Several exoplanet direct imaging instruments will soon be in
operation. They use an extreme adaptive optics (XAO) system to correct the
atmospheric turbulence and provide a highly-corrected beam to a near-infrared
(NIR) coronagraph for starlight suppression. The performance of the coronagraph
is however limited by the non-common path aberrations (NCPA) due to the
differential wavefront errors existing between the visible XAO sensing path and
the NIR science path, leading to residual speckles in the coronagraphic image.
Aims. Several approaches have been developed in the past few years to
accurately calibrate the NCPA, correct the quasi-static speckles and allow the
observation of exoplanets at least 1e6 fainter than their host star. We propose
an approach based on the Zernike phase-contrast method for the measurements of
the NCPA between the optical path seen by the visible XAO wavefront sensor and
that seen by the near-IR coronagraph. Methods. This approach uses a focal plane
phase mask of size {\lambda}/D, where {\lambda} and D denote the wavelength and
the telescope aperture diameter, respectively, to measure the quasi-static
aberrations in the upstream pupil plane by encoding them into intensity
variations in the downstream pupil image. We develop a rigorous formalism,
leading to highly accurate measurement of the NCPA, in a quasi-linear way
during the observation. Results. For a static phase map of standard deviation
44 nm rms at {\lambda} = 1.625 {\mu}m (0.026 {\lambda}), we estimate a possible
reduction of the chromatic NCPA by a factor ranging from 3 to 10 in the
presence of AO residuals compared with the expected performance of a typical
current-generation system. This would allow a reduction of the level of
quasi-static speckles in the detected images by a factor 10 to 100 hence,
correspondingly improving the capacity to observe exoplanets.
