John B. DeVelis

Image Reconstruction with Fraunhofer Holograms

The holograms considered are formed from opaque or transparent diffracting objects which are contained in an aperture illuminated with a coherent, collimated, quasimonochromatic beam of light. It has been shown that the intensity distribution in the near-field of the aperture, but in the far-field of the individual objects which are contained in the aperture, is given by a function whose essential term represents the interference between the Fraunhofer diffraction pattern from the object and the coherent background. The hologram thus formed is referred to as a Fraunhofer or far-field hologram because of the imposed condition. The reconstruction, which is accomplished by placing the recorded hologram in another coherent collimated quasimonochromatic beam and again going to the far-field of the individual objects, yields an intensity which is essentially the original object distribution. In the far-field region of the individual objects for which this result is valid, the reconstruction is seen to be devoid of the evidence of a virtual image. One advantage of this particular method is that the virtual image which appears in the conventional (Fresnel) hologram method creates no problem here since it reduces to a constant for the far-field approximation.

Comparison of Methods for Image Evaluation

The problem of image evaluation for small aberrations by means of the Marechal method of phase-error balancing is well known. For the case of large phase-errors, a number of authors have suggested spot diagram averaging. This method implies an association of ray density with energy flux, which is well justified by experience for large aberrations, but certainly subject to question for small aberrations. Using the radius of gyration as a measure, the ray density method of averaging over the image plane is shown to be equivalent to the simpler and more physical process of averaging over the exit pupil using the radius as a weighting factor. In addition, a proof for the justification of the ray density method is given along with the geometrical intensity law. This proof stems from an application of Liouville’s theorem. Finally, the two limits are compared with results published in the literature for various combinations of aberrations.

Transfer Function for Cascaded Optical Systems

In this paper it is shown that the system transfer function for a cascaded set of optical elements utilizing partially coherent quasimonochromatic radiation is the autoconvolution of an “effective aperture function.” This distribution function is shown to be the product of the aperture distribution functions for each of the cascaded elements of the system. The limiting cases for coherent and incoherent radiation are examined and discussed with respect to their experimental significance.

Effect of Symmetric and Asymmetric Aberrations on the Phase of the Transfer Function

The effect of symmetric and asymmetric aberrations on the phase of the optical transfer function is examined, and it is shown that abrupt phase jumps do not occur; in addition, an unambiguous method of presenting the phase of the transfer function is presented.