QC 351 A7 no. 13 / This paper presents a transfer theory for determination of the image
space and image spectrum of a three-dimensional object. The theory assumes
the existence of volumes of stationarity, called "isotomes," into which
the object must be divided. Isotomicity is approximated, over sufficiently small volumes, in the diffraction- limited case.
The main development assumes the object to be radiating incoherently;
the results are as follows:
The image (irradiance) distribution i(x,y,z) is the three-dimensional convolution of the point spread function s(a,ß,y) with the
object distribution o(x',y',z'). The image spectrum I(wl,w2'w3) is
defined as the three -dimensional Fourier transform of i(x,y,z). It is
found that I obeys a transfer theorem, I = F.0, where F(wl,w2,w3) is the
three -dimensional Fourier transform of s(ct,ß,y) and 0(wl,w2'w3) is an integral
transform of o(x',y',z'). This transfer theorem establishes the value of
using F as a criterion of optical design. In the Fraunhofer approximation,
F may be represented as a line integral across the pupil U. This shows
that F contains a simple pole at col = w2 = O. Nevertheless, all integrals
involving F are convergent. The pupil representation for F also shows
that F is zero outside a restricted volume E of (wl,w2,w3)- space. Because
F is bandwidth- limited, F, I, s, i and T (the optical transfer function)
individually obey sampling theorems. These theorems imply that if each
point in image space is regarded as an independent degree of freedom, there
can be no more than 1/0f4 degrees of freedom/volume in image space.
For coherent object radiation, analogous theorems of convolution,
transfer, and sampling can be constructed. In addition, the "amplitude" transfer function W(wl,w2,w3), defined as the Fourier transform of the
point amplitude distribution u(a,ß,y), is proportional to the pupil func-
tion and to S[w3 - (w2 + w2) /2k], where 6 is the Dirac delta function and
k = 2r/a. This relation is used to establish sampling theorems for u and
for g (the image amplitude) and to express g(x,y,z) as a double integral
over U and O.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/621607 |
| Date | 01 1900 |
| Creators | Frieden, B. Roy |
| Publisher | Optical Sciences Center, University of Arizona (Tucson, Arizona) |
| Source Sets | University of Arizona |
| Language | en_US |
| Detected Language | English |
| Type | Technical Report |
| Rights | Copyright © Arizona Board of Regents |
| Relation | Optical Sciences Technical Report 13 |
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