Image Processing Reference

In-Depth Information

Appendix B: The Phase Retrieval

Problem

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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