Image Processing Reference
Appendix B: The Phase Retrieval
At high frequencies, that is, above a few terahertz, it is increasingly diffi-
culty to measure the phase of a scattered field The inverse scattering meth-
ods described here assume that one can measure both the magnitude and the
phase of the scattered field The (Fourier) phase retrieval problem is an old one
with many different approaches to its solution depending on the nature of the
problem (e.g., compact vs. periodic structures).
Given a 3-D transparent object, interferometry or making a hologram using
the scattered field can be used to image it. Classical interferometric tech-
niques are limited to the determination of 2-D fields since they provide the
path length data for single directions of illumination. Holographic interfer-
ometry makes it possible to obtain an interferogram with multidirectional
illumination and determine the optical path length of “rays” passing through
the object in many different directions. There remains the problem of relating
the refractive index distribution to these data, and the assumption generally
has to be made that rays travelling in straight lines model the situation well.
Some work on ray tracing in refracting objects has been carried out. In any
such procedure there are considerable experimental difficulties, such as the
uncertainty in the fringe order when refractive index gradients change sign
with the object.
One of the few experimental and numerical attempts to determine the
structure of an inhomogeneous scattering object was due to Carter (1983).
To simplify matters a 3-D object, but one having a 1-D scattering potential,
was made from two homogeneous rectangular blocks differing in refractive
index by 0.222 and made sufficiently small such that the first Born approxi-
mation was valid. The complex scattered fields containing the information
required to recover quantitatively the 3-D structure of this object were deter-
mined holographically, following the procedure suggested by Wolf (1970).
Wolf showed that from measurements of the intensity distribution of the field
emerging from an off-axis hologram, one can determine the complex field
down to details of ~9 wavelengths with a reference beam angle of −30°. Carter
scanned a small region of such a hologram around the propagation direction,
converted optical density to effective exposure, and determined the average
fringe period. These data were then used to demodulate digitally the ampli-
tude and phase information carried by the hologram fringes. This procedure
would then have to be repeated for all incident wave directions and should be
repeated to reduce noise. Repeats were performed in the experiment described
here for experimental details for a comparison between the Born and Rytov
approximations applied to these data.
One of the main problems encountered in this approach is the difficulty
in maintaining a precise reference and using a holographic technique while
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