We intend to perform accurate measurements of the magnetic field profile in the Meissner state of an ¿extreme¿ low-k superconductor (In, k = 0.06, k is the Ginzburg-Landau parameter) by polarized neutron reflectivity (PNR). The primary goal is verification of the non-local relation between the current density and the vector potential, first predicted in 1953. The essential features of the non-local effect are the deviation of the field profile from the exponential shape for deeper penetration, and a sign reversal at a certain distance. In spite of the many attempts made to verify these predictions, the problem is still open. Our recent experiments have revealed that the problem can be finally solved by applying the PNR technique for the low-k samples. Detailed knowledge about the penetration filed profile will provide a deeper insight into the electrodynamics of superconductivity.