Effect of Source Size on the Resolution in Fourier-Transform Holography

JOHN B. DEVELIS,* DOMINIC J. RASO, AND GEORGE O. REYNOLDS Technical Operations Incorporated, Burlington, Massachusetts 01804

(Received 16 January 1967)

INDEX HEADINGS: Holography; Resolving power; Source.

THE so-called Fourier-transform hologram process is one in which the Fourier transform of a planar object distribution

is combined with an off-axis reference beam 1–5 arising from either a reference source in the same plane as the object or from an interferometric arrangement. Two kinds of methods for creating point-reference Fourier-transform holograms have been intro­duced.1–3,5 The first kind uses a lens to produce the Fourier trans­form of the object, and the second kind utilizes the Fraunhofer-diffraction region to create the Fourier transform of the object.4'5

In the lens method of forming Fourier-transform holograms, the coherently illuminated planar aperture distribution described by

is Fourier transformed by the lens, where δ (ξ) is a delta function in the object plane ξ D is the distribution of the object, and ξ0 is the separation distance from the delta function to the center of the object. Physically, this aperture distribution corresponds to a pinhole on axis, and the planar object of interest centered off axis, both in a plane one focal length in front of a lens. Upon reconstruction, the intensity distribution on film of either of the real images is given by

where m is the magnification of the system and K is a constant. However, it is emphasized that the delta-function representa­

tion for the reference source in these holograms is an idealiza­tion.2,4,5 The finite size of the source should be taken into account

FIG. 1. Reconstructed images that occurred when a 50-µ reference point was used. (The dc spot has been filtered out.)

in any detailed analysis of resolution. A theoretical description of this system4 shows that the object resolution (spatial frequency) is inversely proportional to the size of the pinhole. The following are some of the experimental results obtained to verify the theory which shows that the resolution is inversely proportional to the size of the reference pinhole.

A test of the resolution capabilities of the Fourier-transform hologram system was performed using a lens. The object was the center section of the standard bar target. The source was a He– Ne gas laser (6328 Å); various-sized pinholes (400, 200, 100, 50 µ) were used for the reference point. The focal length of the lens in both steps was 38 cm. The reconstructed image is shown in Figure 1 for the 50-µ pinhole. (The dc spot was removed in

FIG. 2. Reconstructed images resulting from the (a) 400, (b) 200, (c) 100 and (d) SO µ point reference

FIG. 3. Comparisons of the experimental and theoretical results, giving the resolution as a function of the diameter of the reference point. Also shown are the results obtained when a stop corresponding to the size of the Airy pattern was positioned on the hologram.

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reconstruction by placing a piece of black absorbing tape in the image plane.) Note that the reconstructed images are real but inverted with respect to one another as predicted by Eq. (2). Enlarged reconstructed images resulting from pinhole sizes of 400, 200, 100, and 50µ are shown in Fig. 2. As is seen, the resolu­tion of the reconstructed image improves as the pinhole gets smaller. Figure 3 shows a plot of the theoretical limit of the system4

(inversely proportional to pinhole size) along with the experi­mental data obtained when the 200, 100, and 50 µ pinholes were used. (Since the object did not contain sufficiently coarse resolu­tion elements, the 400 µ data were not available.) Also shown are the results obtained when a stop equal in size the to Airy disk of the pinhole was placed over the center of the hologram. This causes a convolution in the image plane which essentially de­creases the resolution by half, as compared to the theoretical value. The agreement between theory and experiment is quite good.

* AJso with Merrimack College, North Andover, Massachusetts. 1 E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964). 2 A. Vander Lugt, Appl. Opt. 5, 1760 (1966); Proc. IEEE IT-10, 138 (1964). 3 G. W. Stroke, Appl. Phys. Letters 6, 201 (1965). 4 J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965). 5 J. T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 1362 (1966).

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