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Phase determination from image intensity measurements in bright-field optics

The use of complementary semicircular or half-plane apertures in the objective lens back focal plane of an optical system to determine from image intensity measurements the complex image wavefunction is discussed. This method for the solution of the phase problem in optics is restricted to bright-field optics, where the primary wave is allowed to interfere with the scattered wave. It is shown that in this modified bright-field situation, using semicircular (half-plane) apertures rather than the normal circular objective aperture, it is possible to determine for complementary positions of the apertures a Hilbert relation between the real and imaginary parts of the image wavefunction. The initial estimate for the complex wavefunction may be systematically corrected for nonlinear effects occurring in the equations for the image intensity.