Optical phase obtained by analogue Hartley transformation

When the two-dimensional Fourier transformation is performed with a lens the optical amplitude and phase in the output plane represent the complex transform. It can be shown that the two-dimensional Hartley transform is mathematically equivalent to the Fourier transform, but is real valued; amplitude alone fully represents everything. This is significant because ordinary optical detectors do not respond to phase. Here we describe the construction of an optical system in the form of a modified Michelson interferometer which physically demonstrates that it is possible to produce the Hartley transform of a plane luminous object. It is thus possible to encode in the form of amplitude the half of the information in a diffraction pattern that normally is carried in the form of phase.