The aimed high sensitivities and large fields of view of the new generation of interferometers impose to reach high dynamic range of order $sim$1:$10^6$ to 1:$10^8$ in the case of the Square Kilometer Array. The main problem is the calibration and correction of the Direction Dependent Effects (DDE) that can affect the electro-magnetic field (antenna beams, ionosphere, Faraday rotation, etc.). As shown earlier the A-Projection is a fast and accurate algorithm that can potentially correct for any given DDE in the imaging step. With its very wide field of view, low operating frequency ($sim30-250$ MHz), long baselines, and complex station-dependent beam patterns, the Low Frequency Array (LOFAR) is certainly the most complex SKA precursor. In this paper we present a few implementations of A-Projection applied to LOFAR that can deal with non-unitary station beams and non-diagonal Mueller matrices. The algorithm is designed to correct for all the DDE, including individual antenna, projection of the dipoles on the sky, beam forming and ionospheric effects. We describe a few important algorithmic optimizations related to LOFARs architecture allowing us to build a fast imager. Based on simulated datasets we show that A-Projection can give dramatic dynamic range improvement for both phased array beams and ionospheric effects. We will use this algorithm for the construction of the deepest extragalactic surveys, comprising hundreds of days of integration.
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